r\hmS         LIBRARY 

1    /V    j¥^                                     0F  THK 

UNIVERSITY   OF   CALIFORNIA 

• 

Received. C^~4dU2-  *—y 

i8f/ 

Accessions  No.jZjL//^^?     Shelf  No. 

</* 

©A 


AN    INTRODUCTION 


TO   THE 


Logic  of  Algebra. 


Witb  Wlustrative  Exercises* 


BY 


ELLERY  W.  DAVIS,  Ph.D.  (Johns  Hopkins), 
Professor  of  Mathematics  in  the  University  of  South  Carolina. 


NEW  YORK: 

JOHN    WILEY    &    SONS, 

53  East  Tenth  Street. 
1890. 


Copyright,  1890, 

BY 

Ellery  W.  Davis. 


ROBERT  DRTTMMOND,  FERR1S   BR03" 

Electrotyper,  PrinterS> 

m&m  Pearl  Street,  326  Pearl  Street' 

New  York.  New  York. 


ERRATA. 


Rage  viii,  line  30.  For  a  —  (±  b)a  =  q=  b  read  a  —  (±  b) 
=  a  =F  b 
"      xi,  line  3.     For  th$  read  the 


3,  "   1 

7,  " 

24,  u     3- 

29,  "  40. 

38,    "  14. 
40,    "  17. 
49,  diagram. 
52,  /zVz^  12. 


52, 
61, 


14. 
9- 


a  -f-  b  read  b  -f-  a 
a  "      a^ 

(-b      «      (-b) 

55  "56 

of  positive  ra?^  of  a  positive 

fraction  "      fractions 

o  "      0 

thou  sands  read  thousands 

thm  "     them 

—  a1  "      —  a* 

y  -  yi      ^  _     y  -  yi 


ai = 


-—  read  a, 


X  —  Xj  X  —  xx 

62,  diagram.     Leftmost  sloping  line  should  join  —  3I1 
and  1 

69,  /w^y  8  and  16.     iw  integrably  rmaf  integrally 

73,  /zW  6.     /^r  —  2  —  2k  r^W  =  2  —  2k 

78,    "20.       "     11  >  o         "        11  ^>  o 

85,  diagram.      The  dot  for  17  is  out  of  place 

87,  line  26.     For  arrows  read  numbers 

98,    "     17.        "      q1  "      q' 

102,    "     12.        "     T(i  +  ix^  read  T(i  +  i"^) 
106,    "    23.        "     (eki1)  "      (eki!) 

in,  diagram.     F  wanted  at  end  of  sloping  line  from  A 
112,  line  20.    For  &  +  (**  +  **  read  el  +  (2n,r  +  I)i 


b) 


112, 

M 

30. 

M    1   read  1 

U3; 

<< 

18. 

"    i'    "      i 

113, 

<< 

32. 

"    ib'  —  b  read  i(b' 

114, 

U 

23. 

rs                     rb 

115, 

(I 

4- 

If                                                1       IT 

DEDICATION 


To  My  Deceased   Father, 
abe  IReverenD  X.  m.  2>av>te, 

IN  GRATEFUL   RECOGNITION   OF  HIS   CAREFUL  TRAINING, 

HIS    INEXHAUSTIBLE    PATIENCE, 

AND  HIS   SELF-SACRIFICING   EFFORTS  TO    FURTHER 

MY   EDUCATION. 


PREFACE. 


THIS  book  is  precisely  described  by  the  title,  and  is  mainly 
the  outgrowth  of  a  conviction  that  the  logic  of  algebra  is  a  very 
much  neglected  study. 

Partly  because  the  processes  of  algebra  are  simple  and  easily 
taught,  partly  because  both  arithmetic  and  algebra  are  generally 
studied  for  the  sake  of  the  processes  rather  than  for  the  sake  of 
discipline,  the  reasoning  which  underlies  the  processes  has  come 
to  be  very  generally  slurred  over  or  even  absolutely  ignored. 

This  fault  would'  be  somewhat  overcome  if  geometry  were 
taught  before  or  along  with  algebra,  so  that  geometric  illustra- 
tions could  constantly  be  given  of  algebraic  principles.  The 
conceptions  of  geometry  are  less  abstract  and  so  more  easy  to 
grasp  than  those  of  algebra,  and  the  reasoning  of  geometry  is 
correspondingly  more  simple.  Furthermore,  geometry  is  usu- 
ally presented  as  a  fixed  and  settled  science  received  in  all 
its  perfection  from  the  hands  of  the  immortal  Euclid.  The 
student  does  not  learn  one  definition  of  a  triangle  and  then 
unlearn  it  for  another :  is  not  told  that  straightness  is  only  a 
special  case  of  crookedness ;  that  the  inside  of  a  circle  is  only 
the  outside  looked  at  in  a  peculiar  way. 

Algebra,  on  the  other  hand,  shows  everywhere  traces  of  its 
origin  and  development.  Numbers  are,  first,  integral  and  posi- 
tive ;  afterwards,  negative,  fractional,  incommensurable,  im- 
aginary, and  double.  There  is  a  corresponding  series  of  mean- 
ings to  the  words  sum,  difference,  product,  quotient,  power, 
root,  and  logarithm.  Moreover,  these'  extensions  of  meaning 
are  all  more  or  less  arbitrary,  and  some  of  them  at  first  sight 
contradictory.      One  has  constantly  to  discriminate   between 


VI  PREFACE. 

just  what  is  argument  and  just  what  is  definition  and  assump- 
tion on  which  future  argument  is  to  be  based. 

But  these  very  difficulties  make  the  logic  of  algebra  a 
peculiarly  useful  and  invigorating  discipline.  Besides,  there  is 
no  branch  of  mathematics  which  gives  one  so  good  a  general 
insight  into  the  whole  body  of  mathematics.  By  it  the  student 
learns  the  meaning  and  relationship  of  processes  that  he  has 
been  using  more  or  less  blindly,  perceives  the  oneness  of 
mathematical  reasoning  whether  veiled  under  the  name  of 
geometry  or  of  algebra,  and  gets  a  glimpse  of  those  methods 
and  conceptions  on  which  the  whole  of  modern  mathematics 
has  been  built  up. 

In  the  hope  that  the  present  little  book  may  contribute  to 
this  desirable  end  it  is  submitted  to  the  indulgence  of  teachers. 
Only  an  introduction  is  attempted,  because  an  attempt  at  more 
would  defeat  the  very  end  in  view.  Thorough  discussion  of  a 
few  things  better  trains  the  mind  than  a  superficial  treatment 
of  many.  My  only  fear  is  lest  I  shall  have  erred  in  giving 
too  much. 

The  student  is  supposed  to  have  a  knowledge  of  geometry 
and  elementary  algebra.  In  Part  Second,  some  knowledge  of 
trigonometry  and  analytic  geometry  will  be  a  help. 

In  the  preparation  of  the  book  various  sources  have  been 
drawn  upon.     These  are  the  more  important : 

Argand,  Sur  le  maniere  de  representer  quantites  imaginaires. 
Clifford,  Common  Sense  of  the  Exact  Sciences. 
De  Morgan,  Trigonometry  and  Double  Algebra. 

"  Calculus,  Introduction. 

Dirichlet,  Zahlen  Theorie. 
Tannery,  Theorie  des  Fonctions  dune  variable  seule. 

My  thanks  are  particularly  due  to  Prof.  Sloan  of  the  Uni- 
versity of  South  Carolina  for  help  in  revising  manuscript,  and 
to  Mr.  Gustav  Bissing  of  Baltimore  for  many  corrections  and 
fruitful  suggestions. 

University  of  South  Carolina, 
January  I,  1890. 


TABLE  OF  CONTENTS. 


PART   FIRST. 

SIMPLE  NUMBERS. 

I.     The  Direct  Processes  with  Positive  Integers. 

i.  Mathematics  is  characterized  by  deductive  reasoning. 

2.  Reasoning  requires  language.     By  growth  of  language  Arithmetic  becomes 

Algebra. 

3.  The  number  of  objects  in  a  group  is  independent  of  the  order  of  counting. 

4.  Addition  is  commutative:  a -\- b  —  b -f- a. 

5.  Addition  is  associative:  (a  -f  b)  -f-  c  =  a  -J-  (b  -f-  c). 

6.  Multiplication  is  commutative:  a  X  b  =  b  X  a. 

7.  Multiplication  is  associative  for  three  numbers:  {a  X  &)  X  c  =  a  X  (b  X  c). 

8.  The  product  of  any  number  of  factors  is  independent  of  the  order  in  which 

they  are  multiplied  together. 

9.  Multiplication  is  associative  for  any  number  of  factors. 

10.  Involution  grows  out  of  multiplication  as  does  multiplication  out  of  addition. 

11.  Involution  is  non-commutative:  ab^=ba. 

12.  Involution  is  non-associative:  a^^icfif. 

13.  The  direct  processes  are  uniform:  a  —a  with  b  =  b'  requires  a-\-b  =  a'-\-b' , 

ay^b—a'  Xb',  and  ab  =  a'6'. 

14.  Multiplication  takes  precedence  of  involution;  addition,  of  both  multiplica- 

tion and  involution  :  a-\-bXcd  =  a-\-[bX  (cd)\,  a  convention. 

15.  Multiplication  is  distributive  to  addition:  (b-\-c)Xa  =  bXa-\-^X^. 

16.  Addition  is  non-distributive  to  multiplication:  {bXc)-\- a  ^z  {b -\- a)  X  {c-\-a). 

17.  Involution  is  distributive  to  multiplication  when  the  product  is  the  base,  but 

not  when  the  product  is  the  index:  (b  X  c)a  =  baXca,  but  ab  X  ac  qt  abXc. 

18.  Multiplication  is  non-distributive  to  involution:  bc  X  aj=(b  X  a)cXa. 

19.  Involution  is  non-distributive  to  addition:  (b -j- c Y  rfz ba -f- <**,  ab  +  c£  a°-\-ac. 

20.  Addition  is  non-distributive  with  regard  to  involution;  bc -\- a  ^t  (b-\-aY+a. 

21.  The  distributive  law  for  involution  is  (b  X  c)a  =  ba  X  c*.     The  index  laws 

are  <?b  X  ac  =  ab  +  c  and  {ab)e  =  ab*c.  * 

22.  Successive  powerings  can  be  performed  in  any  order:  {ab)c  =  (ac)b. 

vii 


Vlll  TABLE   OF   CONTENTS. 


II.    The  Inverse  Processes  with  Positive  Integers. 

23.  The  inverse  processes  are  the  undoers  of  the  direct  processes. 

24.  Subtraction  undoes  addition:  if  a-\-  b  =  c,  c  —  b  =  a. 

25.  Division  undoes  multiplication:  if  a  X  b  =  c,  c  -r-b  =  a. 

26.  Involution  may  be  undone  in  two  ways. 

27.  Evolution  determines  an  unknown  base:  if  ab  =  c,  a=  |/<r. 

28.  Taking  a  logarithm  determines  an  unknown  index:  if  ab  =c,  b  =  ]oga  c. 

29.  A  whole  series  of  processes  may  be  undone:  if  {a-\-bXc)d  =<?,  a=  y/e-r-c—b. 

30.  The  inverse  processes  are  performed  by  '  guess  and  try.' 

31.  The  inverse  processes  are  non-commutative:  a  —  b=f=b—  a,  a-i-bfi  b-i-  a, 

\/a  rj=  y/b,  logo,  b  -J=  logb  a. 

32.  The   inverse   processes   are   non-associative  :    (a  —  b)  —  c  =f=  a  —  {b  —  c), 

{a  -7-  b)  -f-  c  4-  a  -*-  (b  -¥■  c),    V  Vc^    yf  lo&«  loS*  c  4-  ]oS\oga  b  c 

33.  Parentheses  are  sometimes  avoided  by  conventions  aS  to  the  use  of  the 

symbols  +,  — ,  X,  ■*-,  (    f  \  */,  log. 


III.    Negatives  and  Fractions. 

>  > 

34.  The  inverse  processes  lead  to  new  numbers:  a  —  b  =  c  —  d\ia-\-d=  b-\-c. 

<  < 

35.  To  every  positive  number,  -j-  a,  corresponds  a  negative  number,  —  a. 

36.  A  positive  number  is  a  name  reached  by  counting  on  forward  from  zero;  a 

negative,  one  reached  by  counting  off  backward  from  zero. 

(  dividend  multiplies  )   , 

37.  Multiplying  a  {      divi,or  divides      }  the  quotient. 

>  > 

38.  a-*~b  =  c-i-d\laXd=bXe. 

<  < 

39.  a  ■+■  b  =  -,  a  fraction. 

40.  The  bar  of  a  fraction  is  a  sign  of  inclusion. 

41.  3XSXaXbX(c-\-d)  =  3.$ab(c  -\-  d).     Rules  of  precedence. 

^  ,.         ,.,/■•         ,  ..acaa-\-cc 

42.  Between  fractions  he  other  fractions  forever:  if  -  >  -,  -  >  — — -,>  — . 

•  o      d    o      b-\-  d      d 

43.  The  sum  of  two  integers  is  the  integer  reached  by  counting  from  either  as 

we  would  count  from  zero  to  get  the  other. 

44.  Addition    is    indicated   by  writing   numbers   together   connected  by   their 

proper  signs:  (-f-  a)  -\-  (—  b)  =  a  —  b.     To  subtract  a  number  is  to  add 
its  opposite:  a  —  (±  b)  a=    T  b. 

45.  The  order  of  algebrafc  addition  is  indifferent:  -\-a  —  b=— b-\-a. 

46.  Algebraic  addition  is  both  commutative  and  associative. 

47.  To  subtract  an  addition  and  subtraction  expression  is  to  add  its  opposite. 


TABLE   OF  CON TEN TS.  IX 

48.  In  an  involved  addition  and  subtraction  expression  the  sign  of  any  number 

,    ,       \   positive  I  ..        j  even  ) 
is,  upon  the  whole,   j  f  if  an  1  f  number  of  minus  signs  act 

r  (  negative  )  (  odd  ) 

upon  it. 

49.  Addition  and  subtraction  of  integers  gives  naught  save  integers. 

.  .     S  positive   )  .,   ,       ,         j  the  same  sign  ) 

50.  The  product  of  two  numbers  is  jr  r  if  they  have  1  .       .        f . 
3               r                                                t  negative  >                         <  opposite  signs  ) 

51.  The    prbduct    of    any   number   of    factors   n   of   which    are    negative    is 

{  positive    I   .,      .     j  even  ) 

1  •       (  d  n  is  ) 

(  negative  )  t  odd   J 

52.  Multiplication,  when  negatives   enter,  continues   to   be   commutative  and 

associative. 
_,  .  .    j  positive  I  j  the  same  sign  ) 

53.  The  quotient  of  two  numbers  is  i  .      f  if  they  have  i  .       .        \. 

*  1  negative  >  t  opposite  signs  ) 

54.  Between  any  two  negative  fractions  lie  an  infinite  number  of  other  nega- 

tive fractions. 

55.  To  multiply  by  a  fraction  means  to  multiply  by  the  numerator  and  then 

divide  by  the  denominator. 

, ,         ,  ,         ,,..,,.        product  of  numerators 

56.  The  product  of  any  number  of  fractions  is  the  fraction - : . 

product  of  denominators 

The  operation  is  commutative  and  associative. 

_,      (  multiply  )    ,  ,       .=        j    divide    }   ,      .  .  . 

57.  To  -J     ,.   .,      r  bya  number  is  to  1        ,  .  .     h  by  its  reciprocal. 

(    divide    )  (  multiply  ' 

58.  The  result  of  a  chain  of  multiplications  and  divisions  is  independent  of  the 

order  in  which  they  are  performed. 

59.  Any  number  in  an  involved  multiplication  and  division  expression  is,  upon 

,    ,  {  multiplier  )      .    ,  .        .,  .     .  .  . 

the  whole,  a  )      ..   .  f  of  that  expression  if  it  is  acted  upon  by  an 

(     divisor     ) 

j even  )  .  . 

i  r  number  of  division  signs. 

60.  We  cannot  by  multiplication  and  division  with  positive  integers  and  frac- 

tions get  aught  save  positive  integers  and  fractions. 

61.  The  sum  of  any  number  of  fractions  is  a  new  fraction,  whose  numerator  is 

the  sum  of  all  the  products  obtained  by  multiplying  the  numerator  of 
each  given  fraction  by  the  denominators  of  all  the  other  given  fractions, 
and  whose  denominator  is  the  product  of  all  the  given  denominators. 

■62.  Subtraction  of  fractions  enters  as  did  subtraction  of  integers,  carrying  with 
it  negative  fractions  and  algebraic  addition  and  subtraction  of  fractions. 

°3-  §§  5°>  5T»  52!  53>  h°ld  for  fractions.  To  multiply  by  a  multiplication  and 
division  expression  is  to  divide  by  its  reciprocal. 

64.  With  the  numbers  now  introduced,  addition,  subtraction,  multiplication,  and 

division  are  always  possible. 

65.  A  multiplication  and  division  expression   is   powered  by  distributing  the 

index  of  the  power  over  the  factors  of  the  expression. 
\c      d  <  /d       a        .  .        d 


ay       d  c /d        a        ,  .        d 

66.  h  1    =  -  requires  \  ~  —  ~b  and  loS»  ~  =*• 


X  TABLE   OF  CONTENTS. 

_,  .  (positive   )   .  .  „      (multiplying) 

67.  Powering  by  a  i  .      r  integer  is  repeatedly  \      ..  ...  f  unity  by 

<  negative  >  (     dividing     )  '     ' 

the  base.     Powering  by  zero  is  letting  unity  alone. 

68.  A  fractional  power  is  an  integral  root  of  an  integral  power. 


69.  ac  =  J  a.     If  ( ?  ] 


d       e     ,         <?        c 
f  bf        d 


IV.     Incommensurables. 
70.  Evolution  leads  to  new  expression  numbers:    1^2  is  no  integer  or  fraction. 

>      d,  fa\d  >    f,\b 


'■■mvwm- 


72.  Opposites  and  reciprocals  of  incommensurables  lie  between  the  opposites- 

and  reciprocals,  respectively,  of  the  inclusives  of  the  incommensurables.. 

73.  Taking  logarithms   leads   to   incommensurables:    log2  3  is  no  integer  or 

fraction. 

C  g 

74.  If  ( -  ]    >  4  >  [- )*,  then  loga  -7  lies  between  -  and  £. 

W        J        \bl  I J  d  h 

75.  An  incommensurable  is  a  number  dividing  all  fractions  into  two  sets  A  and 

B,  so  that  any  fraction  from  A  is  less  than  any  from  B,  but  yet  no  frac- 
tion from  A  is  largest,  nor  any  from  B  smallest. 

76.  Two  incommensurables  are  equal  if  their  inclusives  are  equal. 

77.  The  results  of  operating  with  incommensurables  are  hemmed   in   by  the 

results  of  operating  with  their  inclusives. 

78.  As  always,  the  inverse  operations  are  mere  undoers  of  the  direct. 

79.  To  hem  in  all  incommensurables,  we  do  not  need  all  fractions:  decimal 

fractions  are  sufficient. 

80.  Ratio  is  a  general  term  for  all  sorts  of  numbers.     The  sign  '  :  '  is  not  iden- 

tical with  the  sign  '  -5- '. 

V.     Illustrations. 

81.  Algebraic   numbers  and  their  addition  and  subtraction   are  illustrated  by 

steps  and  otherwise.     Negative  numbers  are  sometimes  nonsense. 

82.  The  multiplication  of  algebraic  numbers  is  illustrated  by  a  lever. 

83.  The  illustrations  of  §§  81,  82,  may  be  extended  to  fractions,  but  fractions 

are  sometimes  non-sense. 

84.  Lengths  taken  at  random  are  very  likely  incommensurable.     The  illustra- 

tion of  the  lever  is  extended  to  incommensurables. 

VI.    Growth  and  Rate. 

85.  The  foregoing  sections  comprise  the  main  conceptions  of  elementary  algebra, 

86.  A  number  grows  from  one  value  to  another  by  taking  in  succession  all  in- 

termediate values. 


TABLE   OF  CONTENTS.  XI 

87.  When  y  =  ax,  y  grows  with  x  at  a  uniform  rate  a. 

v'  —  v' 

88.  When^  =  x1,  y  grows  at  varying  rate  2x  compared  to  x.     The  ratio  —, 4j 

has  a  definite  value  only  because  of  the  law  connecting  y  and  x  in  th$. 

y  — y'         y  — y 

growth  of  —. — .  from ,. 

X    —  X  X  —  X 

89.  Whatever  law  connects^  growth  with  x  growth,  the  varying  rate  of  growth 

y  —  y 

of  y  compared  to  x  is -. 

X—  X 

90.  If  y  =  ax,  the  rate  of  y  growth  to  x  growth  when  x  =  o  is  hemmed  in  by 

ah  —  1       .  a~h  —  1 
—  and——-. 

91.  When  a  =  2,  the  above  rate  is  0.693  .... 

rate  of  growth 


92.  The  varying  rate  is  2*  X  0.603  =  y  X  0.693.      The  ratio 

*  growing  number 

is  always  0.693  .... 

93.  Any  number  2*  can  be  reached  by  unity's  growth  at  a  logarithmic  rate  r 

with  regard  to  zero  growing  to  -  X  0.693.    The  logarithmic  rate  r  requires 

the  base  2^  -*-  0.693.     The  logarithmic  rate  unity  requires  the  natural  base 
2n-  0.693  =  e  =  2.71828  .... 

•*■   (I  +  «)"  +  ,>^>(I+«)";(I  +  3"+'>^>(I  +  9"-      When  «  is 

k","(,+3""{+i)' 

95.  Money  put  out  at  simple  and  compound  interest,  respectively,  grows  roughly 

at  uniform  and  logarithmic  rates. 

VII.    Graphs. 

96.  By  a  simple  convention  paired  values  of  x  and  y  determine  a  series  of  points 

(x,  y),  forming  the  graph  of  any  given  relation  between  x  and  y.     The 
graphs  of  y  =  \x  and  y  —  2X. 

y  —  y  y  ~~  v 

97. —,  is  the  slope  of  the  line  joining  (x,  jy)and  (x'f  y').    — —,  is  the  slope 

of  a  graph  at  (x  ,  y). 

98.  If,  on  the  graph  of  y  =  ax,  the  line  through  (x,  y)  and  (x,  y)  cuts  the  line  of 

.r's  where  x  =  k,  then,  so  long  as  x  —  x  =  h,  a  constant,  x  —  k  =  -r , 

ah  —  1 

another  constant.     This  gives  a  geometrical  construction  of  the  logarith- 


and  a  geometrical  interpretation  of  e*  =  ( 1  +-) 


mic  curve 


Xll  TABLE   OF  CONTENTS. 

PART   SECOND. 

DOUBLE  NUMBERS. 

I.     Integral  Double  Numbers  and  the  Simpler  Operations. 

99.  y  —  a1  =  ai\  ai-\-bi  =  (a  -\-b)i;  at  X  H  X  ci  =  —  abci.     The  absolute  value 
of  a  product  of  i  and  non-i  numbers  is  the  product  of  the  absolute  values 

of  its  factors ;  the  product  is  an  ]  .[  number  if  the  number  of  i  factors 

t  non-e  J 

[  ;  and,  in  determining  its  sign,  each  pair  of  i  factors  counts  for 
even  ) 

a  minus  in  addition  to  the  minus  signs  before  the  factors 

100.  Non-*  and  i  numbers  are  object  and  group  numbers.     The  double  number 

a-\-bi  marks  the  ath  object  in  the  bth  group.  Two  double  numbers  are 
equal  if  their  i  and  non-t  parts  are  separately  equal,  and  the  sum  of  two 
double  numbers  is  the  number  reached  by  counting  from  either,  as  we 
would  count  from  zero  to  get  the  other. 

101.  The  product  of  a  -f-  bi  by  c  -f-  di  is  the  ath  number  in  the  <5th  group  of  the 

c  -\-  di  system.  The  groups  of  objects  may  be  rows  of  dots.  The  laws 
for  the  multiplication  of  simple  numbers  hold  for  double  numbers. 
Raising  to  integral  non-4  powers  is  done  by  repeated  multiplication. 

102.  Subtraction  is  the  addition  of  an  opposite,  and  division  is  a  guessing  what 

to  multiply  one  given  double  number  by  to  get  another. 

II.     Non-integral  Double  Numbers  :  Tensors  and  Sorts. 

103.  The  interpolation  of  fractions  and  incommensurables  gives  Argand's  dia- 

gram ,  x  -j-  iy  =  (x,  y). 

104.  Fractional  double  numbers  are  simple  fractions  of  double  integers.    Double 

numbers  may  be  of  the  same,  of  opposite,  and  of  different  sorts. 

105.  (a  +  bi)  ■*-  (c  -f  di)  =  (a  +  bi){c  -  di)  +  (^2  +  d>).       T(c+  di)  =  T(c  -  di) 

=  y?  +  d*. 

106.  Every  double  number  is  the  product  of  a  quantity  and  a  quality  factor. 

107.  If  two  numbers  are  of  different  sorts,  the  tensor  of  their  sum  is  less  than 

the  sum  but  greater  than  the  difference  of  their  tensors. 

(  product  )  .     ,      j  product 

1  \  of  two  numbers  is  the  1 

\     ratio     )  l     ratio 


108.  The  tensor  of  the  i  .       f  of  two  numbers  is  the  j  r  of  their 


tensors.  The  tensor  of  a  non-i  power  of  a  double  number  is  the  non-t 
power  of  the  number's  tensor. 

109.  One   number   lies  between  two  others  if,   and  only  if,   in  some  system 

or  other  the  parts  of  the  one  lie  between  the  parts  of  the  others.  a-\-ib 
lies  between  ax  -f*  ib\  and  a%-\-ik%  if,  and  only  if,  (ax  —  a\a  —  aa)  -|- 
(*,  -  b){b  -  *,)  <  o. 

1 10.  A  number  may  be  such  with  reference  to  two  others  that,  in  all  systems, 

its  parts  lie  between  the  parts  of  the  two  others. 


TABLE   OF  CONTENTS.  Xlll 

III.     Complex  Units  and  Non-z  Powers. 

/i  -\-p  /i  —  / 

in.  Because  tfp  -+-  iq  -  4/  — h  *  y  — — ,  it  is  possible  to  take  of  any 

double  number  a  root  whose  index  is  an  integral  power  of  2. 

112.  Any  complex  unit  between  unity  and  the  doubly  positive   complex  unit 

p -\- qi  can  be  expressed  as  closely  as  one  pleases  by  a  fractional  power 
olp  +  qi. 

113.  All  complex  units  are  hemmed  in  as  closely  as  one  pleases  by  integral 

non-z  powers  of  a  doubly-positive  complex  unit  whose  i  part  is  small. 

114.  The  powers  of  the  doubly-positive  complex  unit  can  be  replaced  by  powers 

of  a  positive  negative  complex  unit. 

115.  Any  complex  unit  is,  as  near  as  one  pleases,  a  fractional  power  of  any 

other  complex  unit.  Conversely,  a  complex  unit  can  be  found  that  shall 
come  as  near  as  one  pleases  to  any  assigned  non-z  power  of  a  double 
number.  A  non-f  power  of  a  non-unit  double  number  is  the  product 
power-of-number's-tensor  X  power-of-complex-unit-of-number's-sort. 

IV.     Growths,  Rates,  and  Amounts. 

116.  A  double  number  grows  by  the  separate  growths  of  its  i  and  non-z  parts. 

Growths  of  double  numbers  are  represented  by  graphs;  uniform  growths, 
by  straight  lines;  varying  growths,  by  curves. 

117.  If  u-\-iv=  {c -\- id)(x -\- iy)  and  c  -f-  id  is  constant,  the  growth  of  u-\-iv 

is  the  same  for  the  c  -f-  id  system  that  the  growth  of  x  -j-  iy  is  for  the 
standard  system. 

118.  If  from  each  of  two  numbers  there  is  a  uniform  growth,  there  will  always 

be  one  and  generally  only  one  number  reached  by  the  growth.  If  more 
than  one,  then  an  infinity  of  numbers  is  reached. 

119.  All  numbers  directly  between  a  -f-  id  and  a  -f-  ib'  are  given  by  /(a  -\-  ib)-\- 

t(d+ib')  where  />  o,  /'  >  o,  and  /+/'  =  1. 

120.  A  single  uniform  growth  from  a-\-  ib  to  c-{- id  is  more  direct  than  a  chain 

of  uniform  growths  from  a-\-ib  to  c-\-id  through  numbers  not  directly 
between  a  -\-  ib  and  c  -f-  id. 

121.  If   the    chain    of    growths    joining  a  -\-  ib   to   c -\- id  through   xx -\- iyx  , 

xi  +  iyi  .   •   •  •   »  *n  -f"  hn »  1S  sucn  tnat  a  <xi  <Xi  <  .  .  .  <xn  <  d  and 

Vi  —  b       y<i  —  Vi  d  —  yn     ,  .,  .... 

> ~  >    .   .   .   >  i—%  then  a  uniform  growth  joining  two 

Xi  —  a       Xi  —  Xi  c  —  xn 

numbers  on  different  growths  of  the  chain  cannot  contain  a  third  num- 
ber on  the  chain. 

122.  A  chain  of  growths  of  the  same  character  as  that  of  §121,  but  through 

numbers  all  directly  between  a-\-ib  and  numbers  on  that  chain,  is  more 
direct  than  that  is. 

y  — y 

123.  Numbers  taken  on  a  varying  growth  from  a  -f-  ib  to  <r-f-^such  that  

x     x 


XIV  TABLE   OF  CONTENTS. 

decreases  with  increasing  x,  determines  two  chains  of  growths  related 
like  those  of  §  122. 

124.  By  taking  numbers  on  the  varying  growth  closer  and  closer  together,  the 

"i  .  f  direct  of  the  two  chains  determined  by  the  two  numbers 
<  less    J 

(     less  and  less     )      .  ■         ,.„ 

becomes  1  ,  c  direct,  and  the  difference  of  the  amounts 

I  more  and  more  ) 

of  the   two  chains  becomes  as  small   as  one   pleases.     Each  amoun 

becomes  the  amount  of  the  varying  growth. 

125.  The  total  amount  of  any  growth  can  be  gotten  by  breaking  it  up  into  parts 

for  which  - increases  with  increasing  x,  decreases  with  increasing  x, 

x  —  x 

or  remains  constant. 

V.     Logarithmic  Growths  and  Double-number  Powers. 

126.  When  unity  grows  through  all  complex  units  around  to  unity  again,  the 

amount  of  growth  is  lit  —  2  X  3.i4i592°5  .... 

iit 

127.  By  non-z  powering  of  i  unity  grows  at  the  logarithmic  rate  — ;  and  by  the 

2 
powering  of  i  t,  at  the  logarithmic  rate  i  with  regard  to  zero  growing 
non-z-ward. 

128.  i  is  reached  by  unity's  growth  at  the  logarithmic  rate  unity  with  regard  to 

iit 
zero  growing  /-ward  to  — .    A  double-number  power  of  a  double  number 

kf-lg-         lf+2  — 

is  a  double  number:  {ekil)f+si  =e         *.*  ». 

(         i\n       I         i\ni 

129.  ( I -| )    =[XH 1     =/*■  when  »  =  00. 

130.  The  numbers  I  +  P-±ll,  L  +  l±^J ',  (,  +  l±^J  ,  ...  are  all  on 

a  growth  from  unity  of  the  logarithmic  sort  p  -j-  qi,  and  the  amount  of 

/         p+m\*      t  [  k±        \ 
growth  from  unity  to  f  1  -\ 1    is  -  ( en  —  1 1.     Geometric  illustra- 
tions. 

131.  Every  number  has  k  distinct  ^th  roots.     All  these  roots  have  the  same 

tensor,  but  no  two  of  them  lie  on  the  same  logarithmic  growth  from 
unity.  An  incommensurable  power  of  a  number  is,  as  near  as  one 
pleases,  any  number  having  the  right  tensor. 

132.  If  a-\-ib  is  a  logarithm  to  base  e  of  c-\-id,  so  also  is  a-\-{2mt-\- 1)6,  where 

n  is  any  integer. 

133.  By  proper  choice  oip-\-iq  the  logarithmic  growth  {p -\-  z'^)-ward  from  one 

of  two  given  numbers  will  contain  the  other. 

134.  The  c-\-id  power  of  a  -\-  ib  is  the  result  of  unity's  growing  at  the  logarith- 

mic rate  \oge  (a  -\-ib)  with  regard  to  zero  growing  (c  -\-  ui)-\va.rd  to  c-\-id. 


TABLE    OF  CONTENTS.  XV 


VI.     Tensor  Representation:  Sines  and  Cosines. 

135.  When  e*  =  r,  ea+ib  —  rb.     A  growth  of  rb  is  determined  by  any  relation 

between  r  and  b. 

136.  If  lb  ~p-\-iq  and    1       _*■    =  p'  -+-  iq  %   any   intermediate   complex  unit, 

2048 

*  +  2048 .  is  very  nearly  mp  +  np'  +  Amq  +  nq')>  where  m  -f-  n  =  1. 

137.  u  =  cos  b-\-i 'sin  £.     Unity  =  570  17'  44". 8. 

138.  Conclusion:  Retrospect  and  prospect. 


-0  !fcv-  A.  .JstJs^ 

'OHITSRSITTI 

AN   INTRODUCTION 

TO  THE 

LOGIC    OF   ALGEBRA. 


Part   First. 

SIMPLE   NUMBERS. 


I.    The  Direct  Operations  with  Positive  Integers. 

1.  The  essential  characteristic  of  mathematics  is  that  all 
truths  therein  are  established  by  reasoning.  Suggested  they 
may  be  by  observation  and  confirmed  by  experiment ;  but 
finally  settled  they  must  be  by  rigid  deduction,  by  showing 
that  the  statements  to  be  proved  are  necessary  consequences 
of  other  statements  already  known  to  be  true  or,  at  any  rate, 
taken  to  be  true. 

When  I  say,  "  All  snow  is  white ;  this  is  snow ;  therefore  this 
is  white,"  the  mental  process  is  so  simple  as  almost  to  escape 
attention.  Yet  it  is  by  the  constant  repetition  of  just  such 
processes  that  the  most  profound  researches  in  mathematics 
are  carried  on.  The  successive  steps  are  easy ;  the  difficulty  lies 
in  seeing  what  steps  to  take. 

2.  Reasoning  must  be  carried  on  by  language  of  some 
sort.  Of  course  the  more  perfect  the  language,  the  more 
clearly  it  brings  before  the  mind  the  statements  needed  and 
their  relations;  and  the  more  completely  it  shuts  out  all  irrele- 


2  AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

vant  ideas,  the  greater  the  ease  will  be  with  which  the  mind 
takes  in  the  reasoning. 

The  intricacy  of  mathematical  reasoning  has  necessitated  a 
language  peculiar  to  that  science.  In  the  symbols  and  notation 
of  arithmetic  we  have  its  beginnings.  When  from  the  consid- 
eration of  particular  numbers  we  pass  to  the  consideration  of 
numbers  in  general,  the  language  grows.  New  terms  are 
added  and  the  meaning  of  the  old  ones  is  enlarged  ;  arithmetic, 
the  art  of  computatio?t,  becomes  algebra,  the  science  of  numerical 
relations. 

3.  We  are  all  familiar  with  the  names  one,  two,  three,  .  .  . 
In  counting  a  group  of  objects  we  apply  to  them,  one  by  one, 
these  names  in  order,  till  each  object  has  its  name.  The  last 
name  given  is  the  number  of  objects  in  the  group.  Manifestly 
this  name  is  independent  of  the  order  of  counting. 

"  Manifestly,"  did  I  say  ?  How  is  it  manifest  ?  Did  any  one 
ever  try  it  for  all  possible  groups  of  objects  ?  Can  any  one  so 
try  it  ?  Do  we  not,  after  all,  assume,  because  it  has  turned  out 
true  in  the  many  cases  in  which  we  and  others  have  tried  it, 
that  therefore  it  must  always  be  true?  The  assumption  is 
natural,  perhaps  justifiable;  nevertheless  it  is  altogether  need- 
less. 

For,  let  k  stand  for  any  number  whatever,  and  /  for  the 
next  greater  number.  Imagine  that  a  certain  group  of  objects 
were  counted  in  all  possible  orders,  and  that  every  count  gave 
k  for  the  number  of  objects  in  the  group.  Add  an  object  to 
the  group  and  count  again.  If  this  new  object  is  counted  after 
all  the  others,  it  takes  the  name  or  number  next  after  k,  that 
is  /.  If  counted  before  some,  then  these  take  each  a  number 
next  greater  than  one  given  them  in  some  previous  counting. 
In  particular,  the  object  then  counted  last  and  so  called  k  is 
still  counted  last,  but  must  now  be  called  /. 

Thus  if,  in  counting  any  group,  the  last  name  or  number 
given  is  independent  of  the  order  of  counting,  it  remains  so 
when  we  add  an  object  to  the  group.  But  starting  with  a 
single  object,  and  adding  objects  one  by  one,  gives  any  group 
whatsoever.     At  the  start  k  is  one  and  /  two  ;  then  k  two  and 


DIRECT  OPERATIONS    WITH  POSITIVE  INTEGERS.  3 

/  three,  k  three  and  /  four,  .  .  .  :  at  first  evidently,  and  so  at 
-each  successive  stage,  and  therefore  finally,  there  is  one  and 
only  one  result  of  counting. 

4.  Suppose  two  groups  of  objects  A  and  B.  Let  the  num- 
ber of  objects  in  A  be  a;  and  in  B,  b.  Count  all  the  objects  in 
both  groups,  beginning  with  the  objects  from  A.  The  last 
object  counted  from  A  takes  the  number  ay  and  so  the  first 
from  B  the  number  a  -\-  1,  a  increased  by  one,  the  next  a  '-\-  2, 
and  so  on  ;  the  last  taking  the  number  a  -\-  b,  a  increased  by  b. 
This  is  the  total  number  of  objects  in  both  groups.  But  count 
first  the  objects  from  b  and  this  same  total  number  is-*— |— £7— 

.  * .     a  -\-  b  =  b  -\-  a. 

The  sum  of  two  numbers  is  independent  of  the  order  of  adding — 
addition  is  commutative. 

5.  Similarly,  if  there  were  a  third  group  C  of  c  objects,  we 
should  find 

a  +  b  -\-c  =  c-\-a-\-b  =  b-\-c-\-a  —  c-\-b-\-a  =  b-\-a-\-c  as  a-\-c-\-b. 

Now  a  -\-  b  -f-  c  means  {a  -\-  b)-\-cy  #-increased-by-£  increased 
by  c,  and  b  -f-  c  -\-  a  means  (b  -f-  c)  -\-  a.  But  this  last  is  the 
-same  as  a  -f-  (b  -f-  c) ; 

...     {a  +  b)JrC  =  a  +  {bJrc)j 

(2  +  3)  +  7  or  5  +  7  =  2  +  (3  +  7)  or  2  +  10 ; 

and  the  sum  of  three  numbers  does  not  depend  at  all  upon 
which  two  of  the  three  were  first  gathered  into  a  partial  sum. 

In  this  proof  we  used  commutation  quite  needlessly.  For 
notice  :  in  the  expression  a  +  b  +  c  we  think  of  the  objects  in 
A  as  named  from  1  to  a,  of  those  in  B  as  named  from  1  to  b, 
and  of  those  in  C  as  named  from  1  to  c.  In  (a  -\-  b)  +  c  the 
objects  in  B  are  named  from  a  -f-  1  to  a  -f-  b  ;  and  in  a  -f-  (b  +  c) 
the  objects  in  C  are  named  from  b -\-  1  to  b -\- c.  But  pass 
from  either  {a  -\-  b)  -\-  c  or  a  +  {b  -f-  c)  to  the  final  sum,  and  the 
objects  in  A  are  named  from  1  to  a,  those  in  B  from  a  +  I  to 
a  -f-  £,  and  those  in  C  from  a-\-  b  -{- 1  to  a  -\-  b  -\-  c.     The  final 


4  AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

result  hangs  only  upon  the  names  finally  given  and  not  at  all 
upon  the  various  changes  of  name  taken  by  the  objects  in  get- 
ting that  result. 

The  same  reasoning  shows  that  the  manner  of  grouping 
does  not  affect  the  sum  of  four,  five,  any  number  of  numbers , 
i.e.,  addition  is  associative. 

The  student  may  prove  that  if  the  associative  law  holds  for 
the  sum  of  k  numbers,  it  must  hold  for  k  -\-  I  numbers  and 
therefore  universally.  Also,  he  may  show  that  the  sum  of  any 
number  of  numbers  is  independent  of  the  order  of  adding. 

6.  Suppose  b  groups  of  a  objects  each.  The  total  number 
of  objects  in  the  groups  is 

a-\-a-\-a-\-a-\-a-\-  .  .  .  to  b  as. 

We  call  this  sum  the  product  of  a  by  b  and  write  it  a  X  b,  a 
multiplied  by  b. 

From  each  of  the  b  groups  take  an  object ;  they  together 
form  a  group  of  b  objects :  another  object  from  each  of  the 
groups,  and  we  have  another  group  of  b  objects.  In  all  there 
are  a  such  groups  of  b  objects  each.  Together  they  contain 
b-\-b-\-b-\-b-\-  .  .  .  to  a  b's  or  b  X  a  objects. 

.-.     a  X  b  =  b  X  a. 


3X4  =  4X3- 

In  words,  the  product  of  two  numbers  is  independent  of  the 
order  of  the  factors — multiplication  is  commutative. 

7.  Just  as  we  understand  a  -\-  b  -f-  c  to  mean  (a  -\-  b)  -f-  c,  so 
we  understand  a  X  b  X  c  to  mean  (a  X  b)  X  c.  This  is  the 
number  of  objects  in  c  groups  of  £-groups-of-#-objects.  In  each 
of  the  c  groups  there  are  b  sub-groups,  and  in  all  the  c  groups 
b  X  c  sub-groups.  As  each  of  these  b  X  c  sub-groups  contains 
a  objects,  the  number  in  all  of  them  is  a  X  (b  X  c). 

.  * .     {a  X  b)  X  c  =  a  X  (b  X  c) ; 


DIRECT  OPERATIONS    WITH  POSITIVE   INTEGERS.  5 

and  for  the  product  of   any  three   numbers,  multiplication  is 
associative. 


E.g., 


(4  X  2)  X  3  or  8  X  3  =  4  X  (2  X  3)  or  4  X  6. 

8.  Because  multiplication  is  both  commutative  and  associa- 
tive, 

a  X  b  X  c  =  bXcXa  =  cXa  Xb  =  cXbXa  =  bXaXc  =  aXcXby 

and  the  product  of  three  numbers  is  independent  of  the  order 
of  the  factors. 

Suppose  that  the  product  of  any  number  of  factors  up  to 
n  inclusive  is  independent  of  the  order  of  the  factors  ;  then  also 
is  the  product  of  n  -\-  I  factors. 

If  there  is  any  change  in  the  final  result,  it  must  be  due  to 
starting  with  different  pairs  of  factors ;  for  the  moment  we  mul- 
tiply together  any  two  of  the  ?i  -\-  i  factors,  we  have  then  to 
deal  with  but  n  factors,  the  pair-product  and  n—  I  other  factors. 

Let  a,  b,  c  be  any  three  out  of  the  n  -f-  I  factors.  Starting 
with  a  X  b,  we  can  go  on  as  we  please  and  so  have  a  x  b  x  c. 
Likewise,  starting  with  a  X  c,  we  can  take  for  the  first  three 
a  X  c  X  b.  But  aXbXc  =  aXcXb,  and  so  the  product  of 
the  n  -|-  i  factors  is  the  same  starting  with  a  X  b  as  starting 
with  aX  c.  If,  however,  we  can  change  one  of  the  factors  in 
the  first  pair,  we  can  the  other  also  ;  i.e.,  the  first  pair  of  factors 
can  be  any  two  out  of  the  n  -f-  I  factors.  Thus  the  proposition 
is  proved. 

But  the  product  of  two  factors,  and  also  that  of  three  fac- 
tors, is  independent  of  the  order  of  the  factors :  then,  too,  is 
that  of  four  factors,  of  five,  six,  any  number  of  factors. 

9.  In  getting  the  final  product  the  factors  can  be  grouped 
into  partial  products  in  any  way  we  please. 

E.g.,     aXbXcXdXeXf=aX{bXcXd)X(eXf). 


6  AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

For  the  expression  on  the  right  is 

(b  X  c  X  d)  X  aX  (e  Xf)  =  b  X  c  X  d  X  a  X  {e  X  f) 
=  (eXf)XbXcXdXa  =  eX/Xl>XdXeXf 
=  aXbXcXdXeXf) 

or,  without  using  the  commutative  law, 

a  X  {b  X  c  X  d)  X  {e  X  f)  =  a  X  [(b  X  c)  X  d]  X  {e  X  f) 
=  aX[bX(cXd)]X(eXf)  =  (aXb)X{cXd)X(eXf) 
=  {aXbXc)xdx{eXf)  =  {aXbXcXd)X{eXf) 
=  (a  X  b  X  c  X  d  X  e)  X  f  =  a  X  b  X  c  X  d  X  e  X  f. 

The  general  proof  of  this,  the  associative  law  for  multipli- 
cation, will  be  an  excellent  exercise  for  the  student. 

10.  We  wrote  for  a  -\-  a  -\-  a  -\-  .  .  .  to  b  as,  a  X  b.  We 
now  write  for  a  X  a  X  a  X  ...  to  ^  as,  ab,  and  call  the  ex- 
pression the  bth.  power  of  a.  a  is  the  base,  b  the  index  of  the 
power,  and  b  is  the  exponent  of  a.  In  getting  the  expression,, 
a  is  said  to  be  raised  to  the  £th  power  or  to  be  powered  by  b> 
and  the  process  is  called  involution. 

11.  Base  and  index  cannot  in  general  be  interchanged. 
E.g.,  23  =  8,  but  32  =  9  ;  25  =  32,  but  52  =  25.  Nevertheless, 
24  =  42  =  16.  By  trying  a  number  of  cases  the  student  can 
probably  satisfy  himself  that  2k  r£  k2  unless  k  =  2  or  4;  then, 
with  slightly  more  difficulty,  that  3*  jt  k3  unless  k  =  3.  Later 
he  may  be  able  to  see  under  just  what  conditions  ab  =  ba.  A 
single  case  of  failure,  however,  suffices  to  show  that  for  involu- 
tion there  is  no  commutative  law. 

On  the  other  hand,  a  thousand  successes,  even  though  we 
had  not  come  upon  a  failure,  would  not  have  proved  the  law. 
They  would  only  create  a  presumption  in  its  favor  and  make 
it  worth  our  while  to  look  further :  to  inquire  into  the  reason 
of  the  successes,  and  see  if  that  reason  must  hold  in  all  cases 
and  so  necessitate  the  law. 

12.  Involution  is  non-associative.  For  2(32)  =  29  =  512,  but 
(23)2  =  82  =  64.    Here,  again,  there  is  not  always  failure.    Thus 


DIRECT  OPERATIONS    WITH  POSITIVE  INTEGERS.  7 

(3>)2  =  92  =  81,   and    3^  =  3<  =  81  ;    (52)2  =  252  =  625,   and 
5<*>  =  54  =  625. 

13.  One  property,  however,  involution  shares  with  addition 
and  multiplication.  All  three  are  uniform  processes.  That  is 
to  say,  the  results  of  the  processes  cannot  be  changed  without 
changing  the  numbers  used  in  getting  the  results.  In  symbols — 
a  =  a'  and  b  =  b'  require  a  +  b  =  a'  +  b',  a  X  b  =  a'  X  b',  and 

a°=  a'6'. 

14.  We  now  consider  expressions  in  which  more  than  one 
of  the  above  fundamental  processes  occur. 

To  avoid  the  useless  writing  of  parentheses  „we  agree  that 
the  parts  of  an  expression  separated  by  +  signs,  the  terms  of 
the  expression,  are  to  be  first  calculated  and  then  the  results 
added.  Addition  is  said  to  take  precedence  of  both  multiplica- 
tion and  involution.     Thus  7  +  2  X  3+ 52  =  7  +  6  +  25  =  38. 

In  the  same  way,  multiplication  takes  precedence  of  involu- 
tion: 3  X  52  =  3  X  25  =  75.  The  multiplications  are  performed 
upon  the  results  of  the  involutions,  the  additions  upon  the  re- 
sults of  the  multiplications ;  hence  the  use  of  the  word  '  prece- 
dence.' 

Whatever  is  connected  with  the  exponent  of  a  power  by 
any  sign  forms  part  of  that  exponent.  ab+c  means  a(b+c);  abXc, 
db™  ;  and  a?,  a™. 

Notice  that  all  the  above  is  arbitrary ;  other  agreements  as 
to  our  use  of  mathematical  language  might  be  made,  sometimes 
have  been  made.  We  merely  follow  prevailing  usage,  a  usage 
that  has  come  about,  as  most  changes  in  language  come  about, 
from  the  attempt  to  express  ideas  with  as  little  trouble  as  pos- 
sible. 

Let  the  student  calculate  the  value  of  these  expressions : 


2  1+1  1  2X3  3 

22X%        2I+I     ,        3%        42+'      X2+5X32- 

15.  Consider  the  expression  b  X  a  +  c  X  a.  It  denotes  the 
number  of  objects  in  a  groups  of  b  objects  each,  together  with 
the  number  of  objects  in  a  groups  of  c  objects  each.     With 


8     AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

each  of  the  groups  of  b  objects  we  can  place  a  group  of  c  ob- 
jects, thus  forming  a  groups  of  (b  -\-  c)  objects.  In  these  there 
are  (b  +  c)  X  a  objects. 

.-.     bXa  +  cXa  =  (b  +  c)Xa. 

Moreover,  since  multiplication  is  commutative, 

aXb  +  aXc  =  aX(b  +  c). 

.         In  the  expressions  on  the  left  the  product  is  said  to  be  dis- 
tributed ;  and  we  say  multiplication  is  distributive  with  regard  to 
addition  whether  the  sum  be  multiplier  or  multiplicand. 
The  proof  is  easily  pictured  to  the  eye,  thus : 


2X3  +  3X3  =  (2  +  3)X3  =  5X3. 
The  student  may  prove  that 

(a  +  b)X(c  +  d)  =  aXc  +  aXd+bXc  +  dXd; 

and,  in  general,  the  product  of  two  sums  of  numbers  is  the  sum 
of  all  the  products  obtained  by  multiplying  each  number  of  the 
one  sum  by  each  number  of  the  other. 

Then  let  him  state  and  prove  a  rule  for  the  product  of  three 
or  more  sums. 

When  from  the  distributed  product  we  pass  back  to  the 
single  term,  from  a  X  b-\-  a  X  c  to  a  X  (b -\- c),  the  terms  a  X  b 
and  a  X  c  are  said  to  be  collected,  summed,  or  added. 

16.  We  have  seen  that  both  addition  and  multiplication  are 
commutative  and  associative.  We  have  just  now  established 
a  relation  between  the  two  processes.  Is  this  relation  recipro- 
cal ?  is  addition  distributive  with  regard  to  multiplication  ?    In 

(b  +  c)  X  a  =  (b  X  a)  +  (c  X  a) 


DIRECT  OPERATIONS    WITH  POSITIVE  INTEGERS.         9 

can  the  signs  '  -f- '  and  *  X  '  be  interchanged  and  the  equality 
hold  ?  (Notice  the  introduction  of  parentheses  above  on  the 
right  and  explain  it.) 

We  have  (b  +  a)  X  (c  +  a)  =  b  X -c  +  a  X  c  +  b  X  a  +  a2; 
but  plainly  (b  X  c)  -\-  a  cannot  exceed  the  first  two  terms  of 
this ;  and  therefore  addition  is  non-distributive  zvith  regard  to 
multiplication,  the  proof  not  being  changed  in  character  when 
the  product  is  the  second  instead  of  the  first  term  of  the  sum. 


3  X  4  +  2  =£  (3  +  2)  X  (4  +  2). 

17.  In  order  to  more  easily  see  the  relations  between  in- 
volution and  the  other  processes,  adopt  for  the  time  being  a 
notation  for  involution  similar  to  that  for  multiplication,  writing 

ab  =  a  p  by 

where  the  p  may  be  read  'powered  by.' 

At  once,  then,  in  order  that  involution  shall  be  completely 
distributive  with  regard  to  multiplication  requires  that  both  in 

(b  +  c)  X  a  =  (b  X  a)+  (c  X  a) 

and  in  a  X  (b  +  c)  =  (a  X  b)  -f  (a  X  c) 

we  shall  be  able,  keeping  the  equality  true,  to  write  p  and  X 
for  X  and  +. 
The  first  gives 

(b  Xc)pa  =  (bpa)  X  (cpa), 
or  (b  x  c)a  =  ba  X  c*, 

a  true  equality  ;  for 

(b  X  c)a  =  (b  X  c)  X  (b  X  c)  X  (b  X  c)  X  .  .  .  to  a  (b  X  c)*s 

=  (b  X  bX  b  X  ...  to  ^  b's)  X(cXcXcX...toa  c's) 
=  baXca. 


IO         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

Hence,  involution  is  distributive  with  regard  to  multiplication 
when  the  product  is  the  base. 

When  the  product  is  the  index  the  distributive  law  would 
require 

a  p  (b  X  c)  =  (a  p  b)  X  (ap  c), 
or  a**'  =  ab  X  ae , 

This,  however,  is  a  false  equality,  for 

ab  X  <f  =  (a  X  a  X  a  X  .  .  .  to  b  as)  X  {aX  a  X  <*X  .  .  .to  c  a's) 

=  a  X  a  X  a  X  .  .  •  to  {b  -f-  c)  a's 
=  ab+c  y±  abXc,  unless  b  =  c  =  2. 

Consequently,  involution  is  non-distributive  with  regard  to  mul- 
tiplication when  the  product  is  the  index. 

18.  In  order  that  multiplication  shall  be  distributive  with 
regard  to  involution  we  need 

(bpc)Xa=(bX  a)p(cXa) 
and  a  X  (b  p  c)  =  (a  X  b)  p  {a  X  c) ; 

that  is  to  say,  we  must  have 

If  x  a  =  {p  X  d)cXa 
and  aX  bc  =  (a  X  b)aXc ; 

both  which  equalities  are  false  unless  a  =  i.     Therefore,  mul- 
tiplication is  non-distributive  with  regard  to  involution. 

19.  Involution   is   non-distributive   with   regard  to  addition 
when  the  sum  is  the  base, 

{a  -\-  c)  p  a  ■£  (b  p  a)  -\-  (c  p  a) 
or  {p  +  cf  ^IfArC*. 

For,  unless  a—  1,  we  shall  get,  on  expanding  (b-\-c)a,  other 
terms  in  addition  to  ba  and  ca . 

Neither  is  it  distributive  when  the  sum  is  the  index, 

ap(b  -\-  c)  :£  apb  -\-  ape, 

ab+c£  ab  +  ac. 


DIRECT  OPERATIONS    WITH  POSITIVE  INTEGERS.        II 

For   we    have  already  seen    that   ab+c  =  a5  X  <^,    and  ab  X  ac 
^t  ah  -\-ac  unless  a  —  2  and  b  =  c  =  i . 

20.  Addition  is  no?i-distributive  with  regard  to  involution. 

&  +  a&{b+ay+at      and      a  +  P  ^(a  +  b)a+c . 
Any  exception? 

21.  Since  #*  X  cf  =  o*+* 

it  follows  that  #*XC  or  a*  +  *+6  +  *+  •  •  •  toc^s  is 

*»  X  <**  X  «*  X  **  X  .  .  .  to  c  ab's  =  (ab)c; 

and  thence,  &<****=;{  [«T]T. 

It  is  because  (#*/  is  expressed  by  abXc  that  we  use  a10  to 
mean  tfw. 

By  §  17,  fjft*  product  of  the  same  powers  of  several  bases  is  the 
product-ofthe-bases  raised  to  the  common  power. 

This  is  the  distributive  law  for  involution. 

Also,  the  product  of  powers  of  a  common  base  is  that  base 
powered  by  the  sum  of  the  given  indices. 

We  now  see  that  a  pozver  is  itself  raised  to  a  power  by  mul- 
tiplying its  index  by  the  index  of  the  power  to  which  it  is  to  be 
raised. 

These  two  are  the  laws  of  the  power  index. 

22.  Though  involution  is  non-commutative,  yet  it  has  a 
property  resembling  commutation.  Using  the  p-notation  and 
agreeing  that  the  operations  denoted  by  a  succession  of  p's  shall 
be  performed  in  order  from  left  to  right,  we  have 

Now,  if  in  these  expressions  we  keep  a  first,  the  other  letters 
may  be  put  in  any  order  we  please ;  for  the  changed  expres- 
sions would,  like  the  original  ones,  all  be 


a  p  {b  X  c  X  d  X  e)  =  a 


yyb  x  c  x  d  x  e 


12         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 


II.    The  Inverse  Operations  with  Positive  Integers. 

23.  Not  only  is  it  true  that  the  three  fundamental  processes 
are  uniform,  giving  with  the  same  numbers  the  same  results ; 
but  also  it  is  true  that  if  either  of  the  numbers  combined  in  the 
operations  is  changed  the  results  will  be  changed.     Thus, 

b  >  c  requires  a  -f-  b  >  a  -\-  c,  a  X  b  >  a  X  c,  ab  >  ac,  ba  >  c*. 

There  is  one  exception  :  1*  =  \c  even  though  b  >  c. 

The  problem,  given  one  of  the  7iumbers  determining  a  result 
of  one  of  the  three  processes,  to  find  the  other  number,  leads  to  the 
four  new  processes  of  subtractiojt,  division,  evolution,  and  taking 
logarithms. 

24.  Subtraction  is  the  process  that  undoes  addition.  Thus,  if 
a  -f-  b  =  c,  then  c  —  b  =  a,  c  diminished  by  b  is  a.  But,  by 
commutation,  b  -{-  a  =  c,  and  so  c  —  a  —  b. 

The  number  after  the  minus  sign  is  said  to  be  subtracted 
from  the  number  before  it.  The  sum  is  now  called  the  minu- 
end, the  number  subtracted  the  subtrahejid,  and  the  result  the 
remainder. 

25.  Division  is  the  process  that  undoes  multiplication.  Ac- 
cordingly, if  a  X  b  =  c,  then  c  ~-  b  =  a;  and  because  of  commu- 
tation, c  -T-  a  =  d.  The  number  before  the  sign  -5-  is  said  to  be 
divided  by  the  number  after  it ;  or,  if  we  prefer,  we  may  say 
that  the  number  after  the  sign  is  divided  out  of  the  number 
before  it.  So  X  is  sometimes  read  multiplied  into.  The  product 
that  we  divide  is  called  the  dividend,  the  number  dividing  the 
divisor,  and  the  result  the  quotient. 

26.  Since  involution  is  non-commutative,  it  may  be  undone 
in  two  ways :  so  as  to  determine  an  unknown  base  or  so  as  to 
determine  an  unknown  index. 

27.  The  undoing  of  involution  that  gives  the  base  is  called 
evolution.  When  ab  =  c,  we  write  fyc  =  a,  and  read,  "  the  bth 
root  of  c  is  a."  c  is  called  the  root-base  and  b  the  root-index. 
The  root-index  2  is  commonly  omitted. 


INVERSE   OPERATIONS    WITH  POSITIVE  INTEGERS.        1 3 

28.  The  undoing  of  involution  that  gives  the  index  is  called 
taking  the  logarithm.  Thus,  ab  =  c  gives  b  =  logac,  b  is  the 
logarithm  to  the  base  a  of  c.  In  the  early  development  of  alge- 
bra this  process  was  overlooked,  and  so  has  come  to  be  classed 
as  non-algebraic  or  transcendental,  the  other  processes  being 
called  algebraic. 

29.  If,  starting  with  any  number,  we  perform  a  series  of 
operations  upon  it  and  thus  get  another  number,  we  can  of 
course  get  back  to  the  starting  number  by  merely  undoing  these 
operations  in  an  order  the  reverse  of  that  in  which  they  were 
performed. 

E.g.,  if  [{a  +  d)XcY  =  e, 

then         a  =  [(  #)-*.  c]  -  b%    b—  [(  pe)  +  c]-a, 

c  =  (  pe)  -r-(a  +  d),    d=  \og[{a+b)xc]e; 

all  which  equations  may  be  verified  by  substituting  numbers  for 
the  letters. 

In  like  manner,  given  the  equalities  below,  let  the  student 
express  each  letter  in  terms  of  the  others. 

[_{[V(^~eXfy]+g\  +hj  =  i- 
(aXb  +  cY=[(e~f)-gf. 

It  will  be  well  for  him  to  test  his  results  by  the  substitution 
of  numbers  for  letters.  He  can  easily  devise  for  himself  as 
many  more  problems  of  this  kind  as  he  pleases. 

30.  The  first  three  processes  that  we  took  up  are  called 
direct ;  the  last  four  are  called  inverse.  In  particular,  evolution 
is  the  first  inverse,  and  taking  a  logarithm  the  second  inverse  of 
involution. 

The  actual  performance  of  the  inverse  operations  is  a  guess- 
ing and  trying  founded  upon  previous  knowledge  of  the  results 
of  the  direct  operations. 


14         AN  INTRODUCTION    TO    THE   LOGIC  OF  ALGEBRA. 

We  know  that  12  —  5  =  7  because  7  +  5  =  12;  that 
$6-1-7  =  8  because  8  X  7  =  56 ;  that  4/144  =  12  because 
122  =e  144 ;  that  log38i  =  4  because  34  =  81. 

Consider  the  division  of  2461  by  23. 

23)2461  (  107  Because  200  X  23  =  4600  >  2461, 

23  and  100  X  23  =  2300  <  2461, 

161  .-.     200 >  2461  -f-  23  >  100. 

161 

Again,  because  2461  —  2300  =  l6l,  and  161  is  less  than 
10  X  23,  .*.  2461  <  no  X  23.  Because  20,  which  is  smaller 
than  23,  would  be  contained  barely  8  times  in  161,  we  guess  7 
for  the  number  of  times  that  23  is  contained  in  161.  Our  guess 
proves  right,  and  consequently  2461  ~  23  =  107. 

After  this  fashion  can  be  analyzed  any  example  in  division. 

In  mathematics,  as  in  other  sciences,  guessing  and  trying 
are  the  two  most  important  tools  that  the  student  has.  Let 
him  not  shrink  from  their  use. 

In  the  following  equations  he  may  guess  values  of  x  and  y 
that  will  make  the  equations  true. 

*2  +  7X^=i8;2X^-3X*2  =  4;3*  =  ^3;  log2  x  =  64  ; 
log,  16  =  2;  5X^+3X;  =  8;  5X^+3X7=19;  3X  x 

+>x/=58;  11  x  x-  (\/y)  =  I2;  *2+y=7\  *'=*2\ 

Vx  =  3 ;  f  =  64 ;   Vx+y  =  3  ;    frx'+y2  =  x  —  y. 

Sometimes  in  the  above  he  may  find  more  than  one  value 
or  set  of  values. 

31.  We  return  to  the  inverse  operations.  Are  they  com- 
mutative ?  Try  it.  We  have  8  —  2  =  6;  12-^4  =  3;  ^8=2; 
log2  16  =  4;  but  2  —  8,  4-5-12,  ^3,  logl6  2,  have  as  yet  no 
meaning.  ~ 

Hence,  the  inverse  operations  are  not  commutative. 

32.  Nor  are  they  associative^  <-Thus :  (8  —  4)  —  2=4— 2  =  2, 
but  8  —  (4  —  2)  =  8  —  2  =  6;  (24  -^  4)  -=-  2  =  6  -r-  2  =  3, 
but    24  -~  (4  -r-  2)  =  24  -f-  2  =  12  ;    \/{  .^256)  =  1/4  =  2,  but 

(V4)/ 

y  256  =  4/256  =  16;     log2  (log4  4096)  =  log2  8=3,  but 
loS(iog2  4)  4096  =  log2  4096  =  64. 


INVERSE   OPERATIONS    WITH  POSITIVE  INTEGERS.        1 5 

The    student    may   have    some   difficulty   in    seeing    that 

|/(  y^)and  V/ c,  loga  Qogbc)  and  log(ioga*)  cy  differ  as  to  form  in 

the  same  way  that  (a  +  b)  -f-  c  and  a  +  {b  +  c)  do.  The  nota- 
tion is  somewhat  confusing.  To  make  matters  plainer,  just  as 
we  wrote  ab  =  apb,  we  will,  for  the  moment,  write  yb  =  b  up  a> 

b  *  unpowered '  by  a ;  then  fy(  yd)  is  (c  up  3)  up  #,  while  \     c  is 


^up(£uptf),  and  the  analogy  is  manifest. 

Again,  if  in  ah  we  regard  b  as  changed  into  a  new  number 
by  writing  a  to  the  left  and  below  it,  we  can  speak  of  b  as 
4  based'  by  a,  and  write  ab  =  bha  ;  with  which  goes  loga  b  = 
b  ub  a,  b'  unbased '  by  a.  So  we  get  loga  (log*  c)  =  {c  ub  b)  ub  a> 
and  log(ioga<5)^  =  ^ub(^ub^),  all  difficulty  vanishing. 

If  the  student  is  unsatisfied  with  the  disproof  just  given  of 
the  commutative  and  associative  laws  for  the  inverse  processes, 
he  will  find  it  not  very  difficult,  after  going  on  a  little  farther, 
to  work  out  a  proof  based  upon  the  definitions  of  the  processes. 
Thus  will  he  show  not  merely  that  the  laws  do  fail,  but  also 
why  they  fail. 

33.  In  the  various  expressions  considered  in  the  immediately 
foregoing  paragraphs,  certain  parentheses  are  rendered  unnec- 
essary by  the  following  conventions  : 

In  expressions  containing  the  signs  -}-,  — ,  X,  -^-,  |/,  log, 
(     )(  \  the  precedence  is — 

1st,  +  -  ;  2d,  X  +  ;  3d,   |/log;  4th,  (     )<  K 

A  chain  of  operations  denoted  by  -f-  and  —  signs  are  per- 
formed in  order  as  if  all  the  signs  were  -f- ;  a  chain  denoted  by 
X  and  -r-  signs  as  if  all  the  signs  were  X  ;  a  chain  denoted  by 
|/  and  log  signs  as  if  all  the  signs  were  \/. 

Since  the  symbols  \/  and  log  are  written  to  the  left  of  the 
numbers  on  which  they  operate,  the  operations  denoted  by  a 
chain  of  them  are  performed  in  order  from  right  to  left. 

Whatever  is  connected  with  a  root-index  or  logarithm-base 
by  any  symbol  whatever  forms  part  of  that  root-index  or  loga- 
rithm-base. 


1 6         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

Thus,  8—4  —  2  is  the  same  as  (8  —  4)  —  2  ;  24  -f-  4  -f-  2, 
as  (244-4)^2;  VV256,  as  4/(^/256);  ^256,  as  (V4^256; 
log2  log4  4096,  as  log2  (log4  4096) ;  logloS2  4  4096,  as  log(log2  4)  4096. 

Again,  log2  ^64  =  log2  8  =  3;  ylog2  16=4/4=2;  log2  4* 
=  log2i6  =  4;(log28)2  =  32  =  9;  ?/26  =  |/64  =  4 ;  (  f/S)2  =  22  =  4. 

The  student  may  remove  unnecessary  parentheses  from  the 
expressions  in  §  29. 

III.    Negatives  and  Fractions. 

34.  We  have  seen  that  in  certain  cases  the  inverse  opera- 
tions failed.  Algebra  takes  up  these  failures,  compares  them, 
reasons  about  them,  operates  with  them, — in  short,  converts 
them  into  numbers.  Thus,  she  "  marshals  victory  out  of 
defeat,"  and  presses  on  to  new  and  ever-widening  fields  of  con- 
quest. 

Consider,  now,  her  methods. 

5  —  8  is  to  be  a  new  sort  of  number  as  are  likewise  6  —  9 
and  6  —  10.  All  three  represent  attempts  to  take  away  more 
units  than  there  are  to  take.  In  5  —  8  and  6  —  9  we  are  asked 
to  take  away  3  too  many;  in  6  —  10,  4  too  many.  We  say 
then 

5—8  =  6  —  9,     but     5  —  8  >  6  —  10. 

To  obtain  a  general  test  of  equality  and  inequality,  suppose 
a,  b,  c,  d  to  be  four  numbers ;  and  first,  let  a  >  b  and  c>  d. 
Then  a  —  b  is  some  ordinary  number,  say  k,  and  c  —  d  is  also 
an  ordinary  number,  say^-; 

i.e.,         a  —  b  =  k,  and     c  —  d  =  g. 

a  —  k-\-b,  and  c=g-\-d; 

whence  a-\-d=k-\-b-\-d,     and     b-\-c=g-\-b-\-d. 

>  > 

Plainly,  a-\-d=  b  +  c  if ,  and  only  if,  k  =  g\  i.e.,  if 
<  < 

a  —  b  =  c  —  d. 
< 


NEGATIVES  AND  FRACTIONS.  If 

Thus,  when  a  —  b  and  c  —  d  are  ordinary  numbers, 
a-\-d=  b  -j-  c   is   the   necessary   and  sufficient  condition    that 

a  —  b=  c  —  d. 

< 

We  impose  the  same  conditions  on  the  new  sort  of  num- 
bers ;  and  thus,  always  a-\-  d=  b  -\-  c  requires  a  —  b  —  c  —  d. 

Let  the  student  prove  that  a  —  b—  c  —  d\ib  —  a  =  d  —  c. 

<  > 

35.  In  particular,  suppose  a  =  b  and  c  =  d;  then  a-\-c  = 
b  -f-  d,  and  so  a  —  b  =  c  —  d.  But  a—  b  is  a—  a,  and  c  —  d is 
c  —  c. 

Hence,  a  —  a  —c  —  c  —  o,  say, 

no  matter  what  two  numbers  a  and  c  may  be.     This  is  a  new 
number,  zero  or  nanglit,  the  symbol  of  nothing  to  count. 

Ordinary  numbers,  1,  2,  3,  4,  ...  ,  may  be  conceived  as 
gotten  by  adding  to  zero,  and  we  may  write  them  o  -j-  i>  O-f-  2, 
o+3>  0  +  4,  .  .  .  ;  or,  for  short,  -f  I,  -f  2,  -f  3,  +4,  .  .  . 
We  call  them  positive  numbers. 

If  a  —  b  =  o  -f-  cy 

then  b  —  a  —  o  —  c  ; 

for  the  test  gives,  writing  b,  a,  o,  and  c,  in  place  of  a,  b,  c,  and 
d,  respectively, 

b  -\-  c  =  o-\-  a  =  a, 

a  direct  consequence  of  a  —  b  =  c. 

Thus,  to  any  positive  number  c  there  corresponds  a  number 
gotten  by  subtracting  c  from  zero.  Furthermore,  the  corre- 
spondence is  reciprocal,  and  all  the  new  numbers  can  be  so 
gotten.  We  write  them  o—  1,  o  —  2,  o  —  3,  o  —  4,  .  .  .  ,  or, 
for  short,  —  1,  —  2,  —  3,  —  4,  .  .  .  We  call  them  negative 
numbers. 

The  number  zero,  neither  positive  nor  negative,  is  the  link 
between  the  two  sorts  of  numbers. 


1 8         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

The  signs  -f-  and  — ,  originally  used  to  denote  addition  and 
subtraction,  are  now  used  to  show  whether  a  number  is  the 
result  of  an  addition  or  a  subtraction  from  zero. 

This  double  signification  of  the  signs  need  not  confuse  us : 
the  distinction  is  merely  one  of  point  of  view.  Precisely  similar 
is  that  between  a -\-  b  regarded  as  a  command  to  add  b  to  <z,  or 
as  the  single  number  resulting  from  the  addition. 

The  value  of  a  number  without  regard  to  its  sign,  or  say 
with  its  sign  taken  positive,  is  called  its  absolute  value  ;  and 
numbers  whose  absolute  value  is  the  same,  but  which  differ  in 
sign,  are  called  opposite  numbers. 

36.  We  now  have  the  series  of  names, 

...  -  4,  -  3,  -  2,  -  1,  o,  +  1,  +  2,  +  3,  +4,  .  .  . 

extending  ad  libitum  on  forward  and  off  backward  from  zero. 

A  positive  number  is  simply  a  name  to  which  we  come 
when  we  count  on  forward  from  zero ;  a  negative  number,  a 
name  to  which  we  come  when  we  count  off  backward  from  zero. 

One  number  in  the  series  is  larger  than  another,  if  we  have 
to  count  off  backward  to  get  from  it  to  the  other ;  and  smaller, 
if  we  have  to  count  on  forward. 

Opposite  numbers  are  equally  removed  opposite  ways  from 
zero,  and  their  absolute  value  is  the  number  of  their  removes 
from  zero. 

37.  In  somewhat  the  same  way  that  we  compared  the  ex- 
pressions 5  —  8,  6  —  9,  and  6  —  10,  we  can  compare  expressions 
like  a  -5-  b. 

When  b  exactly  divides  a,  we  know  that  the  larger  a  is,  b 
remaining  unchanged,  the  larger  is  a-^b;  and,  on  the  other 
hand,  that  when  a  is  unchanged,  the  larger  b,  the  smaller  is 
a  -=-  b. 

Thus,       15 -^  3  >  12  — 3,     and     1 5  -^-  5  <  1 5  -=-  3. 

Further,  multiplying  a  multiplies  #  -=-  b;  while  multiplying 
by  leaving  the  division  exact,  divides  a -i-  b. 

For,  suppose  a-i-  b  =  c,  c  -r-  d  =  e,  and  f  is  any  number ; 
thentfX/-^-  b=fXa  +  b=fx(cXb)  +  b=fXcXb  +  b 


NEGA  TIVES  A ND   FRA  CTIONS.  1 9 

as  /  X  f=/X(«T^  which  proves  the  first  part  of  the  state- 
ment. 

Again,   a  =  b  X  c,  but  c  =  d  X  * ; 

and  so,         a  =  J  X  (^  X  e)  =  e  X  {b  X  <*). 

.  • .     a  -f-  (£  X  ^)  =  e  X  (£  X  <*)  -5-  (£  X  </)  =  e 
=  c  +  d  =  (a  -i-  b)  -i-  d, 

which  proves  the  other  part. 

It  follows  at  once  that  if  k  be  any  number, 

(a  X  k)  -+  (b  X  k)  =  a  -f-  b ; 
for,  (*  X  £)  -*■  (^  X  k)  =  a  X  k  -^  (£  X  3)       , 

^aXk-:rk-^b  =  a^rb. 

38.  Suppose  two  expressions  a  -f-  b  and  c  -±  d.  Moreover, 
let  the  division  be  exact. 

By  what  we  have  just  proved, 

a  -*-  b  =  (a  X  d)  -$-  (£  X  </), 

and  c  +  d  =  (bX  c)  +  (bX  d). 

Necessarily,  then, 

a  ^-  b  =  c  -r-  d,  if  a  X  d  —  b  X  c. 
<  < 

This,  the  ^\y/  of  equality  when  division  is  exact,  we  make 
the  definition  of  equality  when  division  is  not  exact.  Thus  the 
test  becomes  universal. 

39.  The  new  sort  of  numbers  whose  equality  we  have  just 
defined  are  fractions,  and  we  write 

b 
where  a  is  the  numerator  and  b  the  denominator  of  the  fraction. 


4/4 

2 
"  11 

log28 

"  3  X  4  "  i 

'      32 

/3f 

2 

=  4/16: 

=  4; 

162 

log3— - 

20         ,4yV  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

In  contradistinction  the  numbers  with  which  we  have  pre- 
viously been  dealing  are  whole  numbers  or  integers. 

4.0.  Whatever  is  connected  with  the  numerator  or  denomi- 
nator of  a  fraction  by  any  symbol  whatever  forms  part  of  that 
numerator  or  denominator,  the  bar  of  the  fraction  serving  as  a 
sign  of  inclusion. 

3  12 

«-;4  +  7  =  4+s« 

=  log,  81  =4;  J=J- 

Thus  powering  is  the  only  operation  upon  a  fraction  for  which 
a  sign  of  inclusion  is  needed. 

41.  Along  with  the  new  division  notation  it  is  convenient 
to  bring  in  an  abbreviated  multiplication  notation. 

Instead  of  X  we  write  .  ,  or  even  merely  write  our  letters 
and  expressions  together. 

Thus,  2X3X5X«X^XH^)  =  2.3.5^(f  +  4 

Notice,  in    passing,  that   3$  means  3  -f-  f,  but  a-  means 

a  X  —.     In  arithmetic  -f-,  in  algebra  X,  is  omitted. 

When  multiplication  is  denoted  by  mere  writing  together 
of  the  factors,  all  operations,  save  only  involution  and  evolu- 
tion, take  precedence  of  it. 

Accordingly, 

24  -T-  2  .  3  =  24  -7-2X  3  =  36 ;  but  abc  -r-  be  =  a; 
log,  ab  =  log,  (ab) ;  log,  a{b  +  c)  =  log,  \a{b  +  c)]. 

There  is  division  of  usage  as  to  the  meaning  of  \/ab,  some 
making  it  Vab,  others  \/a  .  b  =  b  \/a.  The  latter  usage  is  more 
general. 


NEGATIVES  AND  FRACTIONS.  21 

If  the  first  be  adopted,  fya  fyb  ought  to  mean  Va  \/b  ;  if  the 
other,  tya  tyb  would  mean  tya  X  tyb. 

So  log,  a  log,  b  ought  to  mean  log,  (a  log,  b) ;  and  for  the 
product  of  two  logarithms,  we  should  write  log,  a  .  log,  b. 

4.2.  If  t  and  -=  are  two  fractions  such  that  ■=-  >  -5.  then  the 
^  o  a  o       a 

fraction  „    ,     ?.  whose  numerator  is  the  sum  of  their  numerators 
b-\-d 

and  whose  denominator  is  the  sum  of  their  denominators,  is 

intermediate  in  value  to  the  two  fractions ; 

a      a-\-  c      c 

Le*'  b>y+d>d' 

For,  7-  >  -i  gives  ad >  be,  whence  ab  -{-  ad>  ab  -f-  be;  or,  what 
is  the  same  thing, 

a(b  -\-  d)>  b{a  -f-  c),      and  so        -r  >  >   ,     ,. 

But  #*/  >  fo  also  requires 

(a  +  <?K  >  ip  +  ^V,    and  hence     ,   .     .  >  -z. 

2(1  +  ^  1.  #         ,  *  +  £ 

In  like  manner,      .    ,  ->  lies   between  7-  and   ,    '      ;  while 
2b-\-d  b  b-\-d 

a  +  2c  ,.      .  #-[-<;  £ 

■7— r ;  lies  between  t—. — ■>  and  -> 

b-\-2d  b-\-d  d 

Plainly,  we  can  go  on  forever  finding  intermediate  fractions. 
In  other  words,  between  any  two  unequal  fractions  lie  an  in- 
finite number  of  other  fractions. 

Find  by  the  above  method  all  the  fractions  of  which  neither 
the  numerators  nor  the  denominators  exceed  10,  and  which  lie 
between  y1^  and  10.  Arrange  them  in  the  order  of  their  mag- 
nitude. 

Given   a   set   of   fractions,    prove   that   any    new   fraction 


22         AN  INTRODUCTION  TO    THE  LOGIC  OF  ALGEBRA. 

sum  of  multiples  of  numerators    . 

t rrr- , ?— ] = — 1 —  ls  smaller  than  the  greatest 

sum  of  multiples  of  denominators  s 

and  larger  than  the  least  fraction  of  the  set. 

If  all  the  fractions  of  the  set  are  equal,  how  about  the  value 
of  the  new  fraction  ? 

The  positive  integers  1,  2,  3,  4,  .  .  .  are  fractions  whose 
denominators  happen  to  exactly  divide  their  numerators.  In 
particular,  then,  between  any  two  positive  integers  lie  an  infinite 
number  of  fractions. 

Analogy  suggests  that  we  ought  also  to  have  fractions  lying 
between  negative  integers.  We  shall  presently  see  how  these 
arise. 

43.  We  now  extend  the  meaning  of  the  words  '  addition  * 
and  '  sum '  so  as  to  apply  them  to  negative  integers. 

The  sum  of  two  integers   is  the  integer  gotten  by  counting 

j   on  forward  \  ''  T  r  (on  forward  ) 

1    xc  z.     z.         ,  X  from  either  as  we  would  count  \     rr  ,     .         .r 
(  off  backward  )J  {.off  backward ) 

from  zero  to  get  the  other. 

To  get  the  sum  of  several  integers,  we  take  the  sum  of  any 
two  of  them,  the  sum  of  that  sum  and  a  third,  of  that  and  a 
fourth,  and  so  on — until  all  of  the  integers  have  been  used. 

Addition  is  the  process  of  getting  this  sum.  Subtraction,  as 
before,  is,  now  and  always,  the  process  that  undoes  addition. 

44.  It  follows  directly  from  the  definition  that  we  can  indi- 
cate the  addition  of  integers  by  merely  writing  them  in  any 
order  connected  by  their  proper  signs. 

E.g.,  the  sum  of  —  7,  +  4,  —  2,  and  -f-  1  is 

-7  +  4  —  2+1,     or    4  —  7+1—2,     or     1  —  2  +  4—7. 

We  do  not  say  that  these  sums  are  the  same;  we  do  not 
say  that  these  are  the  only  sums ;  we  merely  say  that  either  of 
the  above  three  expressions  could,  in  perfect  accordance  with 
our  definition,  be  the  sum  of  the  four  given  numbers. 

Since  to  count  from  an  integer  to  zero  requires  the  same 
number  of  counts  and  in  the  same  direction  as  to  count  from 
zero  to  the  opposite  of  the  number,  it  follows  that  to  subtract 
an  integer  is  the  same  as  to  add  its  opposite. 


NEGATIVES  AND  FRACTIONS.  23 

45.  When  all  the  numbers  added  happen  to  be  positive, 
addition  as  just  defined  falls  in  with  ordinary  addition.  When 
necessary  to  distinguish  it  from  ordinary  addition  we  call  it 
algebraic,  and  the  sum  is  an  algebraic  sum. 

With  algebraic  as  with  ordinary  or  arithmetical  addition,  the 
order  of  adding  is  indifferent. 

For,  consider  the  sum 

a  —  b  —  c  +  d  —  e +/. 
This  is 

1  +  1+1  +  ...  +  1,— 1  —  1  —  1  —  1— ...  —  !,— 1  —  1  —  1  —  ...  —  I,... 

or,         a  counts  forward,  b  backward,  c  backward,  .  .  .  , 

and  we  count 

1,2,  3,  . . .  a—  i, a;  a—i,a—2,  .  . .  a—b,a—b—i,  . . .  a—b—c,  . . . 

The  final  count  is  in  nowise  changed  if  we  remove  the  counts 
a,  a—  1  underlined  above.  This  is  the  same  as  if,  in  the  unit 
additions,  we  removed  the  underlined  combination  -f-  1  —  1,  at 
the  first  change  of  sign.  Of  course  the  combinations  — |—  1  —  1 
and  —  1  — |—  1  can  be  removed  wherever  they  occur.  To  keep 
doing  so  will  finally  get  the  signs  all  of  one  kind.  The  number 
of  units  left  will  be  the  excess  of  the  number  of  units  of  one 
kind  over  the  number  of  units  of  the  opposite  kind.  But  this 
excess  depends  merely  upon  the  number  of  units  of  each  kind 
there  were  at  the  start  and  not  at  all  upon  their  order. 

Hence,  the  sum  of  any  number  of  integers  has  for  its  absolute 
value  the  difference  between  the  sum  of  the  positive  integers  and 
the  sum  of  the  opposite s  of  the  negative  integers  ;  and  for  its  sign, 
the  sign  of  those  integers  that  gave  the  larger  sum. 

45.  Algebraic  addition  is  thus  both  commutative  and  associa- 
tive:  commutative,  because  commutation  is  but  a  special  case 
of  change  of  order ;  associative,  because  by  successive  changes 
of  order  as  we  add  we  can  get  any  desired  grouping. 


24         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

Let  the  student  compare  §  9  and  show  how,  by  changes  of 
order,  to  get 

[(+  «)  +  (-*+(-')]  + 1  (+  d)  +  [(-  *) +(-/)  +  (+/)] r ; 

from    (+a)  +  {-t)  +  (-t:)  +  (+d)+{^e)  +  (- /)  +  {+£% 

46.  From  the  associative  law  it  follows  that  the  sum  of  two 
or  more  addition  and  subtraction  expressions  is  the  sum  of  the 
numbers  entering  into  the  expressions. 

E.g.,      (_«  +  *_, _rf)  +  (_,-f*_/_^) 

=  —  a-\-  b  —  c  —  d—  e-{-  k  —  f  —  g. 

47.  If  all  the  numbers  in  one  addition  and  subtraction  ex- 
pression are  the  opposites  of  the  numbers  in  another  addition  and 
subtraction  expression,  the  expressio?is  are  called  opposites  of  each 
other.  Plainly,  their  values,  i.e.  the  single  numbers  to  which 
they  are  reducible,  are  opposite.  But  to  subtract  a  number  is 
to  add  its  opposite ;  therefore,  to  subtract  an  addition  and  sub- 
traction expression  is  to  add  its  opposite. 

E.g.,      (a-b  +  c-d-e)-(-f-g+£-l) 
=  a  —  b  +  c  —  d-e  +  f+g—k  +  l. 

48.  The  expressions  considered  may  themselves  be  sums  of 
expressions  and  these  in  turn  sums  of  others,  and  so  on. 


E.g.,  a-\b+[c+{f-g-k)-\\. 

This  means  that  the  number  g  —  k  is  to  be  taken  from  f 
the  result  added  to  c,  that  result  added  to  b,  and  this  last  result 
subtracted  from  a. 

By  §§46,  47, 

=  a  —  b  —  c—  (/  —  g  —  k)  =  a  —  b  —  c  —  f-\-g  —  k 
—  a  —  b  —  c  —f+g  —  k. 


NEGATIVES  AND  FRACTIONS.  2$ 

Of  course,  we  could  have  just  as  well  removed  the  signs  of 
inclusion,  beginning  with  the  inmost  and  proceeding  outwards. 

But  the  final  result  can  be  written  without  going  through 
the  intermediate  steps.  For  notice  :  each  number  is  acted  upon 
not  only  by  the  sign  immediately  before  it,  but  also  by  the  sign 
of  each  and  every  expression  in  which  it  is  included. 

Thus  k  is  to  be  counted  backward  from  g,  the  reverse  of 
backward,  i.e.  forward,  from/,  forward  from  c,  forward  from  b, 
backward  from  a.     Hence  its  final  sign  is  — . 

Similarly,  we  can  get  the  signs  of  all  of  the  other  numbers. 

In  general,  every  minus  sign  acting  upon  a  number  reverses 
the  number ,  and  hence  the  sign  of  the  number  is  finally  -f-  or  — 
according  as  an  even  or  an  odd  number  of  minus  signs  act  upon 
the  number. 

The  student  will  do  well  to  build  up  complicated  addition 
and  subtraction  expressions  and  simplify  them  by  the  foregoing 
rule.  He  can  test  his  work  by  substituting  actual  numbers  for 
the  letters  in  both  the  original  expression  and  the  final  simpli- 
fication.    The  calculated  value  of  each  should  be  the  same. 

49.  Evidently  any  addition  and  subtraction  expression  is  by 
the  above  processes  reducible  to  a  positive  or  a  negative  in- 
teger ;  and  we  cannot,  therefore,  by  addition  and  subtraction  of 
integers  get  aught  save  integers. 

50.  Let  a,  by  and  d  be  positive  integers,  and  let  a  -\-  b  =  c. 

Then  cd  —  (a  -f  b)d  =  ad  -f-  bd, 

and  ad  =  cd  —  bd ; 

but  a  =  c  —  b,     and     ad  =  (c  —  b)d; 

,-.(c-  b)d=cd-  bd. 

In  words,  multiplication  is  distributive  with  regard  to  subtrac- 
tion if  the  result  of  the  subtraction  is  positive. 

We  have  not  defined  multiplication  when  negative  numbers 
enter.  Let  us  assume  it  such  that  this  law  still  holds.  Then, 
just  as  (c  —  b)d  =  cd  —  bd,  we  have 

(b  —  c)d  =  bd  —  cd,  or  (—  d)d  =  —  ad. 


26         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

The  product  of  a  negative  number  by  a  positive  is  a  negative 
number  ;  and  by  commutation,  a  positive  number  multiplied  by  a 
negative  likewise  gives  a  negative  result. 

Further, 

(_  a){-  d)  =  (b-  c)(-  d)  =  b{-  d)  -  c{-  d) 
=  —  bd  -j-  cd  =  cd  —  bd  =  ad. 

A  negative  number  multiplied  by  a  negative  gives  a  positive 
number. 

It  goes  without  saying  that,  in  all  cases,  the  absolute  value 
of  the  product  is  the  product  of  the  absolute  values  of  the 
factors. 

51.  In  like  manner,  the  continued  product  of  any  number  of 
positive  and  of  negative  integers  has  for  its  absolute  value  the 
product  of  the  absolute  values  of  the  integers,  and  its  sign  will  be 
-f-  or  —  according  as  the  number  of  negative  integers  is  even  or 
odd. 

52.  Since  the  absolute  value  of  the  product  is  the  same 
when  negatives  enter  as  when  they  do  not,  and  since  the  sign 
of  the  product  depends  only  upon  the  number  of  negatives 
entering,  it  follows  that  the  multiplication  can  be  performed 
in  any  order.  In  other  words,  the  commutative  and  associative 
laws  still  hold. 

53.  Let  ad  =  c;  where  a,  d,  and  c  are  positive  integers. 

Then  (—  d)d  —  a(—  d)  =  —  ad '=  —  c, 

and  (—  a){—  d)  —  ad—  c. 

But  division  is  the  undoer  of  multiplication : 

.  • .     (—  c)  -5-  d  =  c  -S d  =  —  a  =  —(c  -i-d), 

and  (—  c)-i d=a  =  c-^-d; 

c  —  c  c  — c       c 

—d=-d  =~d'  and  —d=i- 


NEGATIVES  AND  FRACTIONS.  2? 

These  equations,  true  when  division  is  exact,  we  assume 
true  when  division  is  inexact,  thus  defining  the  negative  frac- 
tions to  which  reference  has  been  made. 

With  fractions  as  with  integers,  opposites  have  the  same 
absolute  values  but  opposite  signs. 

We  say  that  two  negative  fractions  are  equal  if  the  opposite 
positive  fractions  are  equal ; 

a  _        c  a  _  c 

~b~~df    lf     b~d' 


ct      c 
Hence  t  =  -3  involves — 
o       a 


—  a       —  c 

—  a       c       a       —  c       —  a       —  c 

'ITi,  =  ^d> 

-  b~  d'    b  ~  —d'      b    ~   d  ' 

—  a          c 

a          —  c        a            c 

b     ~  -dy 

~b=~d~'     ZTfi^ZTd' 

Notice  that  just  as  t  =  -j  requires  ad  =  be,  so  in  all  the 

other  fraction  equalities  the  cross-products  are  equal. 

54.  We  know  that  a  >  b   involves  —  a  <  —  b.     We   now 

a        c     ,     11  .         ,  a  C        T»  1 

agree  that  t>  ~j  shall  involve  —  -r  <  — -  -3.     But  the  necessary 

and  sufficient  condition  for  -=-  =-z  is  ad  =  be.     The  correspond- 
ing condition,  therefore,  with  negative  fractions  is 

—  t=—-}>     "     —  ad  =  —  be. 
b  >       d  > 

It  easily  follows  that  if  a  fraction  lies  between  two  others 
its  opposite  lies  between  the  opposites  of  the  two  others.  Con- 
sequently between  any  two  negative  fractions  there  lie  an  infi- 
nite number  of  other  negative  fractions. 

55.  We  return  to  positive  fractions.  In  §  37  we  saw  that  if 
a  -T-  b  was  an  integer,  then  to  multiply  a  by  any  number  or  to 


28         AN  INTRODUCTION   TO    THE  LOGIC   OF  ALGEBRA. 

divide  b  by  any  factor  of  b  would  multiply  a  -r-  b  by  that  num- 
ber or  factor.     E.g., 

24  -f-  6  =  4,     and     (24  X  2)  -f-  6  =  24  -f-  (6  -v-  2)  =  8  =  4  X  2. 

We  assume  this  true  of  all  division  expressions,  i.e.  of  all 
fractions. 

Thus,  no  matter  what  positive  integers  a,  b  and  c  are, 


a  ac 

7-  X  c  = 


b  ~  b       b  +  c1 

and,  if  the  commutative  law  is  to  hold,       L^tn sAAy CutiM' w    J  j/ 


^  X  t  =  ~r  =  c  X  a  -±  b. 


In  words,  to  multiply  by  a  fraction  means  to  multiply  by  the 
numerator  and  then  divide  by  the  denomi?iator. 

1  #       1  I  C  CCl        •  r      ,1 

Since  not  only  c  X  j  but   also  j  X  a  gives  -% ,  it  follows 

that  cXj-=c-Z-bXa,  and  so  we  can  effect  multiplication  by 

a  fraction  by  first  dividing  by  the  numerator  and  then  multi- 
plying by  the  denominator. 

a 
In  particular,  j  is  either  1  X  a  -f-  b  or  1  -f-  b  X  a,  is  either  a 

part  of  a  multiple  of  unity  or  a  multiple  of  a  part  of  unity. 

56.  Immediately  from  the  definition  of  multiplication  by  a 
fraction,  we  get 

a       c    _  ac 
bXd=zJd; 

and  the  product  of  any  number   of  fractions  is   the   fraction 
product  of  numerators 
product  of  denominators' 

Plainly,  the  operation  is  both  commutative  and  associative. 


NEGATIVES  AND   FRACTIONS.  29 

57.  Because  division   undoes  multiplication,  to  divide  by  a 

fraction  is  to  multiply  by  the  denominator  and  then  divide  by  the 

numerator. 

a 
Thus,  to  divide  by  j  is  to  multiply  by  b  and  then  divide  by 

a ;  but  this  is  the  same  as  to  multiply  by  -. 

Now  -7-  X  -  =  —7  =  I.     Two  such  numbers  whose  product 

is  positive  unity  we  call  reciprocals  of  each  other.     For  instance, 

the  multiples  of  unity  2,  3,  4,  5,   ...  ,  are  reciprocals  of  the 

parts  of  unity  £,  -J,  \,  |,  .  .  . 

{  multiply  \  .  (    divJ?    ) 

We  can  now  say,  ^  1      ,«   .,      r  by  a  number  is  to  i        .  .   .    r 

J        (    divide    )    y  (  multiply ) 

by  the  reciprocal  number. 

58.  Having  a  chain  of  multiplications  and  divisions  to  per- 
form, we  can  turn  the  divisions  into  multiplications-by-recip- 
rocals; §  55  then  tells  us  that  the  operations  can  be  performed 
in  any  order.  They  could  therefore  previous  to  the  change  of 
the  divisions  into  multiplications.  The  student  may  give 
examples. 

59.  Consider  the  expression 


Here    g+k=f,   /+jt%=f+f=/x±  =  £; 
<x  </+■*+*>*&■',   »X&X  (/+**?)]  «=*p 

and  finally, 

=  a^-b-+c-±fXg-Jrk. 


i 


30         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

This  result  we  could  have  foreseen  by  noticing  that  k  is  a 
divisor  of  gy  a  divisor  of/'s  divisor  or  a  multiplier  of  f,  a  mul- 
tiplier of  c,  a  multiplier  of  b ;  and  since  b  is  a  divisor  of  #,  k  is 
thus  a  divisor  of  a.  And  any  number  in  the  expression  is  a 
multiplier  or  a  divisor  of  a  according  as  it  is  affected  by  an 
even  or  an  odd  number  of  division  signs. 

Of  course  our  manner  of  indicating  division  has  nothing  to 
do  with  this  result. 

T,        .         a  s  divides  r\  multiplies  f>  divides 

c  d,  multiplies  k,  divides  m,  multi- 

m  plies   c,    divides   b,    multiplies  a. 

d  Likewise  r  divides,  f  multiplies, 

r     d  divides,  k  multiplies,  m  divides, 

J  '   s1    c  multiplies,  and  b  divides  a. 

ackfs 


the  expression  is 


bmdr ' 


The  student  should  practise  himself  in  reductions  similar 
to  the  above  until  he  can  perform  them  with  ease  and  certainty. 
Let  him  compare  §48. 

60.  The  method  of  reduction  just  exhibited  can  plainly  be 
applied  to  all  multiplication  and  division  expressions.  Plainly 
also,  any  chain  of  multiplications  and  divisions  is  indicated  by 
some  such  expression.  We  cannot,  therefore,  by  the  multipli- 
cation and  division  of  positive  integers  and  fractions  get  aught 
save  positive  integers  and  fractions. 

61.  The  addition  of  fractions  has  not  yet  been  defined. 
We  assume  it  to  be  such  a  process  that  multiplication  is  dis- 
tributive with  reference  to  it.  Then,  if  -r  and  -3  are  two  posi. 
tive  fractions,  we  have  by  this  convention 


(y  +  2)  Xbd=ad+bc; 


,   -  a   ,  c       ad  +  be 

and  thence  r  +  ~j  = rh — . 

0        d  bd 


NEGATIVES  AND   FRACTIONS.  3 1 

In  the  same  way, 

_  axbj>%  ■  .  .  bn  +  bxajft  .  .  .  bn  +  .  .  .-  +  bxb%  .  .  .  £,_,*,. 

bxbj>%  .  .  .  bn 

Thus,  the  sum  of  any  number  of  fractions  is  the  fraction 
whose  numerator  is  the  stem  of  the  products  gotten  by  multiplying 
the  numerator  of  each  given  fraction  by  the  denominators  of  all 
the  other  given  fractions,  and  whose  denominator  is  the  product 
of  all  the  given  denominators. 

Evidently  the  commutative  and  associative  laws  hold  as 
with  integers. 

r  a    .   c        e     . 

02.  Just  as  a  -\-  b  =  c  gave  c  —  o  =  a,  so  7  + 1  =  7  gives 

e       c       a 

-z  —  -j  =  r.     Thus  subtraction  of  fractions  enters  as  did  sub- 

j       a       o 

traction  of  integers,  then  negative  fractions,  and  algebraic  addi- 
tion and  subtraction  of  fractions.  Accordingly,  the  sum  of 
*         c  e         1  g . 

*'-2'-/'and*ls 

a        c       e       g      adfh  —  befh  —  bdeh  -f-  bdfg 
b~d~f  +  li=z  Jdffi  * 

The  student  may  show  that  the  definition  of  a  negative 
fraction  just  suggested  agrees  with  that  of  §  53. 

63.  We  can  now  in  §§  50,  51,  52,  53  remove  the  restriction, 
express  or  implied  throughout,  that  the  letters  should  stand  for 
positive  integers,  and  allow  them  to  stand  for  fractions  as  well. 
Thus  we  easily  show  that  the  absolute  value  of  any  multiplication 
and  division  expression  is  the  same  as  if  all  the  numbers  entering 
therein  were  positive,  while  the  sign  is  positive  or  negative  accord- 
ing as  there  are  an  even  or  an  odd  number  of  minus  signs  affecting 
the  factors,  direct  or  reciprocal,  of  the  expression. 

Here,  by  a  direct  factor  we  mean  one  that  multiplies  the 
final  value  of  the  expression ;  by  a  reciprocal  factor,  one  that 
divides  the  final  value  of  the  expression. 


32         AN  INTRODUCTION-   TO    THE   LOGIC  OF  ALGEBRA. 

If  in  any  multiplication  and  division  expression  we  change  all 
the  factors  into  their  reciprocals,  the  resulting  expression  is  the 
reciprocal  of  the  original  expression,  and  to  multiply  by  either 
expression  is  the  same  as  to  divide  by  the  other.  The  student 
may  give  examples.     Compare  §47. 

Can  two  expressions  be  both  opposites  and  reciprocals  ? 

If  I  change  all  the  factors  of  a  multiplication  and  division 
expression  into  their  opposites,  will  the  resulting  expression  be 
the  opposite  of  the  original  expression  ? 

64.  Any  expression  that  can  be  built  up  by  additions,  sub- 
tractions, multiplications,  and  divisions,  let  the  grouping  be 
ever  so  intricate  and  involved,  can,  by  the  mere  performance 
of  the  indicated  operations,  be  reduced  to  a  simple  positive  or 
negative  fraction.  Furthermore,  all  the  rules  applying  to  the 
addition  and  subtraction,  the  multiplication  and  division  of 
positive  integers,  apply  to  these  more  complicated  expressions ; 
for  they  apply  to  the  equivalents  of  these  expressions,  the  sim- 
ple fractions. 

65.  Raising  of  negatives  and  fractions  to  positive  integral 

powers  requires  no  explanation.     Evidently,  ■!  >  powers  of 

(  positive  )  -        - 

negatives  are  \  .      \  ;  and  any  power  of  a  fraction  =  the 

*  (.  negative  ) 

power  of  numerator 

new  fraction ■=— ■ : — ; — . 

J  power  of  denominator 

Likewise   any   multiplication    and    division    expression    is 

powered  by  a  positive  integer  when  all  the  factors,  direct  and 

reciprocal,  of  the  expression  are  so  powered.     In  other  words, 

we  power  the  expression  by  distributing  the  index  of  the  power 

over  the  factors  of  the  expression. 

66.  Since  evolution  and  taking  logarithms  are  the  inverses 
of  involution, 


S)' 


d  \     ,    *  fd      a        .  .       c 

—  requires  both  \/  "  =  t,  and  loga  -7  =  c; 


a  c 

where  7-  and  ~  must  agree  in  sign  if  c  is  odd. 


NEGATIVES  AND   FRACTIONS.  33 

67.  Consider  the  expression  ab  -f-  ac,  where  b  and  c  are  posi- 
tive integers.     By  our  definitions  the  expression  is 

(a  XaXaXaX   .  .  .  to  b  as)-±{a  XaXaXaX  .  .  .  cds). 

Assuming  b  >  c  reduces  the  above  to 

\XaXaXaXaX   •  .  .  to  (b  —  c)  a's  =  ah'c ; 

while  b  —  c  gives  unity,  and  b  <  c  gives 

i-i-a-^-a-^-a-7-a-r-  .  .  .  to  (c  —  b)  a's. 

The  agreement  that  the  last  two  results  as  well  as  the  first 
shall  be  denoted  by  ab~c  extends  our  notion  of  powering,  and 
gives  us  these  definitions. 

Powering  by  a  positive  index  is  repeatedly  multiplying  unity 
by  the  base. 

Powering  by  zero  is  leaving  unity  alone. 

Powering  by  a  negative  index  is  repeatedly  dividing  unity  by 
the  base. 

In  all  cases  the  absolute  value  of  the  index  is  the  number  of 
times  that  the  base  operates  upon  unity. 


Thus, 


=  2-Ky.4?  =  2".32=  18432. 

Express  as  a  simple  fraction  without  negative  indices 

/  q-2  +  a*b-4y2      fa26~ly3  §   /b~h^ 
\a-*b>  -=-  a>b~6)     X  \a~^J      l   WJ    ' 

Also,  write  an  equivalent  expression  without  denominators  or 
the  signs  -^  . 

68.  Before  considering  the  meaning  of  fractional  indices, 
the  student  should  prove  these  equalities.  We  assume  that 
the  indicated  roots  can  always  be  taken.  Thus  the  second 
equality  below  is  to  be  understood :  "  If  there  is  a  ^th  root  of 
a  and  a/th  root  of  that  ^th  root,  and  if  there  is  also  a  ^th  root 


34         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

of  a  and  a  /th  root  of  that  #th  root,  then  the/th  root  of  the 
#th  root  is  the  same  as  the  ^th  root  of  the/th  root,  and  each  is 
the/^th  root  of  a." 

The  product  of  the  pxh.  root,  the  #th  root,  and  the  rth  root 
of  a  is  what  root  of  what  power  of  al 

The  /th  root  of  a  to  the  rth  power,  multiplied  by  the  ^th 
root  of  a  to  the  s  power,  is  what  root  of  what  power  of  a  ? 

To  define  powering  by  a  fraction,  suppose  that  (ab)c  =  abc 
holds  just  the  same  when  b  is  a  fraction  as  when  it  is  an  integer. 

Then  \w)  =  ab,    and  so    &  ==  fyab. 

i 
In  particular,      a?  ==  |/#. 

Let  the  student  now  express  all  of  the  foregoing  equalities 

b  d  b_      d 

in  the  new  notation.     He  can  then  prove  that  ac  X  a*  =  ac    e , 

b  b_  b_ 

and  that  &  X  dc  =  {ad)c. 

b_  c_ 

69.  If  ac  =  df  then  d*>  =  #.     But,  if  we  use  a  fractional  root- 
b_  *    I  c_        b  / 

index,  ac  =  d gives  /  /  d  =  a;  and  so  db  =  /  / d. 

e 

Of  course,  if  I  -A    =  -7,  logo.  -7  =  ->     Find  ^  below. 
L°g§  -V-  =  *>     loSi  x  =  32,     log*  64  =  f . 


INCOMMENSURABLES.  35 


IV.      INCOMMENSURABLES. 

70.    Evolution    introduces   new   expression-numbers.     For 

example,  \/2  is  such  a  number.     Obviously,  the  square  of  no 

integer  is   2  ;    for   (±  i)2  =  1,   (±  2)2  =  4,   (±  3)2  =  9,  .  .  .  , 

numbers   all   different,  from   2.     Neither  is  there  a   fraction 

a        .     , 
7-  such  that 

For  this  requires  a2  =  2b2. 
Is  a  odd  and  =  2^  -["  r>  say? 

Then    <z2  =  (2^  +  i)(2/£  +  1)  =  4^  +  4^  +  I  =  an    odd 
number. 

But  2b2  is  an  even  number. 


(?)•-■ 


.-.  a2  =£  2^2  unless  <z  is  even. 

Suppose,  then,  a  is  even  and  =  2c, 

It  follows  that  a2  =  4c2  =  2#2,  and  so  ^2  ==  2d2. 

This  requires  £  even,  say  b  =  2d;  and  so  c2  =  2a?2. 

Just  as,  at  the  start,  a  was  shown  to  be  even,  we  can  now 
show  c  to  be  even.  But  c  is  the  result  of  dividing  a  by  2. 
Consequently  a  -f-  2  is  even  ;  and  going  on,  c  -f-  2  or  4  -5-  2  -5-  2 
is  even,  and  likewise  a  -f-  2  -5-  2  -r-  2,  a  -5-  2  -4-  2  -5-  2  -f-  2,  .  .  . 
But  every  division  gives  a  smaller  number,  and  so  exact  division 
must  cease  either  when  we  come  to  the  smallest  integer  1,  or 
before  that ;  that  is,  some  time  or  other  we  get  the  square  of 
an  odd  number  equal  to  an  even  number,  a  proved  impossi- 
bility.    Therefore  |/2  is  not  a  fraction. 

In  like  manner  it  can  be  shown  that  1/3  =  7-  requires 
p2  =  if,  where  p  =  3/  ±  I ; 
i.e.,  3?P±P  =  3f,  or  i(lp  ± p)  qF  1  =  if, 

a  manifest  absurdity,  /,  p,  and  q  being  integers. 

%K   OF  THE  ^ 


$6         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

Let  the  student  show  also  that  \/2,  |/2,  .  .  .  |/2,  are  neither 
integers  nor  fractions. 

Later  he  can  show  that  all  powers  of  fractions  are  frac- 
tions, and  that  consequently  if  an  integer  has  no  integral  root 
it  can  have  no  fractional  one. 

71.  The  new  numbers  are  surds  or  irrationals  and  belong  to 
the  large  class  of  numbers  called  incommerisurables  because  not 
having  a  common  measure  with  unity. 

To  define  the  equality  and  inequality  of  surds,  consider  how 
the  equality  and  inequality  of  expressions  has  so  far  been  tested- 

We  had         a  —  b  =  c  —  d,  if  a  -4-  d  =  c  +  b, 
<  < 

and  a  -~  b  =  c  -r-  d,  if  a  X  d  =  c  X  b  ; 

<:  < 

where  of  course  a,  by  c,  and  d  are  supposed  positive  integers. 
We  ought  to  expect,  then,  using  the  p  and  up  notation  of 

§17,  aupb  =  cupd,  if  apd=  cpb) 


or,  in  ordinary  notation, 


\/a  =,  pcy  if  ad  =  **. 


When  we  can  actually  take  the  bth.  root  of  a  and  the  d\h. 
root  of  c,  the  condition  certainly  applies :  for  suppose  fya  =  k, 
and  pc  as  /;  then  ad  =  hbd  and  c1  s±  ldb,  and  plainly 

kbd=  lbd,  if  k%l, 

which  is  precisely  our  condition.     The  condition   is  fulfilled, 
moreover,  even  when,  in  place  of  a  and  c,  we  write  fractions 

-  and  -z.     If  we  now  make  this  test  universal  we  shall  but  be 


INCOMMENSURABLES.  37 

following  our  previous  methods.     Thus,  for  surds  as  well  as 
exact  roots,  we  write 

</"W'r«(")W- 

If  b  as  d,  this  becomes 

b/a  >    b  I c    ,.a>  c 

Thus,  we  say 

4/2  >  i,     1.4,     1.41,     1.414,     1.4142, 
and  \/2<2,     1.5,     1.42,     1.415,     1.4143; 

because  the  squares  of  the  first  set  of  numbers, 

1,     1.96,     1 .988 1,     1.999396,     1 .99996164, 
are  all  less  than  2 :  while  the  squares  of  the  secona  set, 
4,     2.25,     2.0164,     2.002225,     2.00024449, 


-  and  "2  above  may  themselves  be  powers  of  fractions.    Say 


are  all  greater  than  2. 

—  and  ~p  above  ma 
'         / 

they  are  \jj    and  [-]  .     This  gives 

(*)5te)J.'"-'»*t«"-- 


38         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

Arrange  the  expressions  in  the  following  groups  in  the  order 
of  their  magnitudes : 

Hi    W>    if*;    Vh    fa    W; 
(i)f,   (I)4,   (*)s;    1/4,    K    I* 

Prove  that  (t)   >  i,ii  a>  &  and  c>  d\  and  that  otherwise 

fjj   =i.    What  conditions  give  W\  =  i  ?    How  can  #**  =  #*a? 

Show  that  (|)v  =  ft7-)1. 

72.  We  saw  in  §  42  that  between  any  two  positive  fractions 
were  an  infinite  number  of  other  fractions.  Hence  between 
any  two  squares  of  fractions  are  an  infinity  of  other  squares  of 
fractions,  between  any  two  cubes  an  infinity  of  other  cubes, 
and  so  on.  It  follows  that  there  are  always  two  fractions  as 
close  together  as  we  please  between  which  any  required  root 
of  positive  integer  or  fraction  must  lie. 

A  root  lying  between  positives  we  naturally  call  positive, 
and  we  define  its  opposite,  negative  of  course,  as  the  number 
lying  between  the  opposites  of  the  including  positives. 

In  the  same  way,  we  say  that  the  reciprocal  of  a  surd  is  the 
number  lying  between  the  reciprocals  of  the  including  numbers. 

73.  Taking  logarithms  leads  to  incommensurables.  Thus 
there  is  no  integer  or  fraction  log2  3. 

For,  suppose  log2  3  =  ^  ; 


then  2a  =  3*. 

Now  2  divides  2a,  and  therefore  it  divides  3*  or  3 .  3*-1. 
This  makes  it  divide  3*"1;  for  otherwise  there  woula  be  a  re- 
mainder of  I  ;  and  in  the  three  3*"1  's  a  remainder  of  3,  which 
2  does  not  divide.  In  like  manner,  2  should  divide  36-2,  3*"3, 
.  .  .  ,  34>  33>  32>  3>  an  absurdity. 


INCOMMENSURABLES.  39 

Consequently  2  does  not  divide  3*  and  2a  ^  3*,  nor  is  there 
a  fraction  j  such  that  log2  3  =  j. 

Again,  logIO  2,  log10  3,  logIO  4,  logIO  6,  .  .  .  ,  are  incommen- 
surables.  For  all  powers  of  10  end  in  a  cipher,  but  no  powers 
of  2,  3,  4,  5,  or  6  do. 

74.  Whether  or  no  these  incommensurables  can  be  expressed 
as  surds  need  not  now  concern  us.     We  can  treat  them   as 

a  c 

we  did  the  surds.     Thus,  if  2b  >  3  and  2^  <  3,  we  say  that 

a       ,  c 

J>l°g>3>^ 

More  generally, 


«  © 


>/    and    Xd)  </' 


e  c  g 

then       log*  -7.  lies  between  -3  «and  -r. 

When         ^-  >  unity,  j  >  log«y  >  j. 

a  c  e      g 

When        j  <  unity,  j  <  loga  j  <j. 

c  g 

Since  between  any  two  fractions  -*  and  -7  there  are  an  in- 
finite number  of  other  fractions  as  close  together  as  you  please  ; 
and  since  if  a  fraction  lies  between  two  others  the  result  of 
powering  by  that  fraction  will  lie  between  the  results  of  power- 

£ 

ing  by  the  including  fractions  ;  it  follows  that  loga  ^  is  hemmed 

bJ 
in  as  closely  as  you  please. 

Let  the  student  put  the  above  conditions  in  the  b  and  ub 
notation  of  §  17. 

75.  Notice  that  for  both  sorts  of  incommensurables  we  have 
all  fractions  divided  into  two  sets,  such  that  all  in  one  set  are 


40    AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

less  than  any  in  the  other  set,  and  so  that,  moreover,  there  is 
no  largest  fraction  in  the  set  of  smaller  ones,  nor  smallest  frac- 
tion in  the  set  of  larger  ones. 

Were  there  a  largest  fraction  r  in  the  set  of  smaller  ones, 

then  no  fraction  could  be  ever  so  little  larger  than  it  without 

falling  into  the  set  of  larger  ones,  and  the  incommensurable  a 

hemmed  in  between  the  two  sets  could  not  be  ever  so  little 

a 
larger  nor  the  least  bit  smaller  than  jr.     In  other  words,  the 

incommensurable  is  the  commensurable  j,  a  palpable  contra- 
diction. 

By  this  property  we  define  incommensurables. 

An  incommensurable ;  we  say,  is  a  number   that  divides  all 

ef- 
fractions into  two  sets  A  and  B  such  that  any  fraction  —  from 

b  a 

A  is  less  than  any  fraction  t>  from  B,  but  yet  no  —  is  largest, 

nor  aiiy  p  smallest. 

These  incommensurables  we  denote  by  Greek  letters,  and 

a    b 
agree,  if  a  is  an  incommensurable  and  -  „  77  any  fraction  what- 
ever from  the  two  sets  of  limiting  fractions,  that 

a  b 

-J  <a<  17- 
a  o 

It  follows  immediately  that  the  opposites  and  reciprocals  of 
incommensurables,  as  defined  (§  74),  are  themselves  incommen- 
stirable. 

76.  Two  incommensurables  a  and  fi  are  called  equal  if  every 
fraction  smaller  than  a  is  also  smaller  than  ft  and  every  fraction 
larger  than  a  is  also  larger  than  (3 :  are  equal,  in  brief  if  their 
inclusives  are  equal.  Notice  that  the  same  definition  applies  to 
commensurables. 


INCOMMENSUKABLES.  41 

77.  If  two  incommensurables  a  and  ft  are  connected  by  addi- 
tion, multiplication,  or  powering,  we  agree  that  the  results  shall 
be  hemmed  in  by  the  results  of  the  same  operations  on  the  inclu- 
sives  of  a  arid  ft. 


a  c  e  p* 

Thus,  11  o  <  j  <  a  <  2*  and  o  <  -?  <  ft  <  ~, 


a    ,    e  c    ;   g    a       e  eg 

-+7<a+fi<7+£,  ?X7<«X/»<5xf. 


e  S 

while  if  t-  >  1,  (t-J    <  op  <  f-j)  ,  taking  positive  roots  only. 

Of  course  the  inequalities  are  supposed  to  hold  if  in  place 
of  either  a  or  /?  we  have  a  commensurable,  and  everybody 
knows  that  they  must  hold  when  in  place  of  both  a  and  ft  we 
write  commensurables. 

Under  the  same  conditions  as  above  write  the  inclusives  for 
a  —  ft,  a  X  —  fi,  and  a~P. 

Show  that  the  sum  of  a  commensurable  and  an  incommen- 
surable is  incommensurable,  as  is  likewise  the  product. 

Show  that  the  sum  of  two  incommensurables  may  be  com- 
mensurable, as  may  likewise  the  product. 

Show  that  the  primary  operations  with  incommensurables 
are  uniform,  and  that  the  commutative,  associative,  distributive, 
and  index  laws  hold  as  with  commensurables. 

The  last  problem  is  especially  simple.     For  example, 


|-+ i  <*  +  /J<£;  +  f  requires^  +  f  <  ft  +  a  <l  +  C-. 


78.  Since  subtraction  is  the  addition  of  an  opposite ; 
division,  multiplication  by  a  reciprocal ;  and  evolution,  raising 
to  a  reciprocal  power, — we  need  not  specially  consider  these 
processes. 


42         AN  INTRODUCTION    TO    THE   LOGIC  OF  ALGEBRA. 

As  for  taking  logarithms,   loga  /?  =  y  if  ofl  =  /?«      More 

2  f  .  a  c 

explicity,  if   ab  <  «Y  <  a*,  then  y  lies  between  j-  and  -3. 

The  student  may  show  that 

l0ga  fiy    5=  l0ga  /?  +  lOga   7  J    l°g<x   ~    =   l°ga   fi  ~  l0ga  y  J 

loga  /5Y  =  y  loga  /? ;  loga  VP-=\  loga  /? ; 

loga  P  log^  7  =  loga  y  ;    loga  /?  log7  6  —  loga  d  logv  fi  =  logay  08 ; 

ioga  /?  =  i  -T-  log^ « ;  «% Y  =  7% a ;  «loga7  =  y  ; 

log^2  8  ^2  =  ?  log2V3  12  =  ?  logI  +  V2  (3  +  2  4/2)  =  ? 


loga  4/3 . 3^7^2  f^  •    V^  =  ?  l°g|  •  ~7^  =  ? 


2  4/2 

3V3 


79.  By  definition,  -=-  <  \/2  <  -j  if  -r-  <  2  <  -7-.    But  also,  by 

definition,  7-  <  4/2  <  -7  requires  t2-  <  (  \/2)2  <  -7-  .  Conse- 
quently (  \/2)2  =  2.  Similarly,  (  tya)H  =  a  =  ny/an  whether  fa 
be  commensurable  or  incommensurable. 

As  between  any  two  fractions  there  are  an  infinite  number 
of  fractions,  so  between  any  two  exact  integral  nth  powers  of 
two  fractions  there  are  an  infinite  number  of  ;zth  powers  of  frac- 
tions infinitely  close  together.  These  can  be  used  to  hem  in 
incommensurables  without  the  help  of  fractions  not  perfect  nth 
powers.      In    fact,    howsoever    close    together    the    fractions 

j-  and  -j  enclosing   the    incommensurable    a   may   be   taken 


( 7-  <  a  <  -J,  there  are  an  infinite  number  of  fractions  ^  such 


e 

7 


a       r    _  c 
thatJ<7-„<^. 


INCOMMENS  URA  BL  E  S.  43 

These  fractions  are  those  from  the  inclusives  of  a/j  an<* 

a  j-    that  are  greater  than  a/j    and  less  than  a  /-  .     Of 

these  an  infinite  number  have  their  «th  powers  less  than  a  and 
an  infinite  number  their  wth  powers  greater  than  a ;  for  other- 
wise a  would  fall  in  either  with  one  of  the  wth  powers  or  else 

with  t  or  -,,  and  could  not  be  incommensurable. 
o        a 


It  is  tacitly  assumed  above  that  a     -j  and  a 


c 

~j  are  incom- 
mensurable. All  that  would  happen,  if  they  were  not,  is  that 
one  or  both  of  them  would  fall  in  with  nth  powers'  of  the  frac- 

e 
tions  -7,  and  the  reasoning  would  not  be  invalidated. 

In  like  manner,  any  series  of  fractions  that  are  not  limited 
in  size  and  between  any  two  of  which,  howsoever  close  together,  there 
lies  a  fraction  and  so  an  infinity  of  fractions  of  the  series,  may 
be  used  to  hem  in  incommensiirables. 

In  particular,  we  may  use  decimal  fractions. 

The  condition  "  not  limited  in  size"  is  important.  If,  for 
instance,  no  fractions  of  the  series  were  larger  than  10,  the 
series  could  not  be  used  to  hem  in  an  incommensurable  larger 
than  10.  If,  again,  all  fractions  of  the  series  were  either  larger 
than  10  or  smaller  than  5,  no  incommensurable  between  5  and 
10  could  be  hemmed  in  by  the  series. 

An  interesting  application  of  the  above  principles  is  afforded 
in  the  formal  proof  that  ffypn  =  ( Vi/P)n>  Let  "typ*  =  a  and 
'fyp  =  fi ;  we  are  to  prove  that  a  =  fin. 


a  c  a  c 

Wehave   j-  <  a  <  ^      if     ^-</-<-; 


e  JT  em  P"w 

and  7<^<|'      if    ^<P<J^' 


44         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 
Now  4  <  /3  <  jt       involves     jn  <  /?"  <  |- ; 

em  o-m  emn  g.mit 

while  j7n<P<%      involves     -j^-n<pn  <  j^n 

But  j^-n    and    ^--    are  7/zth  powers  of    y-n    and    ^. 

Consequently  /3n  as  well  as  a  lies  between   numbers  whose 

en 
mth  powers  hem  in  pn .     The  numbers  -^-  are  thus  among  the 

a  p~n  c 

numbers  -r,  and  the  numbers  -^  among  the  numbers  -7,  and 

a  =  J3n. 

a 

80.  Consider  the  expression  t>,  where  a  and  p  are  two  posi- 
tive incommensurables.  Let  o  <  a  <  a  <  a'  and  o<b  <  fi  <  b', 
where  the  new  letters  may  be  fractions.     Then 

a       a      a' 
J,<P<J' 

Now  —  may  be  either  integral,  fractional,  or  incommen- 
surable.     Thus,    if    a  =  2/?,    a  <  2<£'    and    a'  >  2$,    so    that 

a  a'       .      .     ..         2  '    <?       2       a'       ,  -      .. 

y<2  <  -j.    Again,  if  or  =  -/?,£,-<-<  -^  ;  while  if  or  =  /?  j/2, 

F  <♦*<»• 

Remove  the  restriction  that  or  and  /?  shall  be  positive,  and 

—  stands  for  all  the  sorts  of  numbers  with  which  we  have  had 

to  do.     For  this  expression  we  have  a   name.     We  call  it  a 
ratio  and  define  it  as  the  number,  be  it  positive,  negative,  in- 
tegral, fractional,  or  incommensurable,  by  which  we  must  mul- 
ct 
tiply  one  number  to  get  another.     The  ratio  -  of  a  to  ft  is 


ILL  US  TRA  TIONS.  4 5 

frequently  denoted  by  a  :  /?,  where  the  sign  :  takes  precedence 
of  all  other  symbols  of  operation. 

E.g.,         a  +  /3Xy:#  +  e-Z  =  -j11— . 

e  ""  ^ 

There  is  another  unique  convention  about  the  use  of  the 


We  write         a  l  b  l  c  :  d  =  k  l  1 1  m  l  n 

to  mean  that  the  ratio  of  any  two  numbers  on  the  left  is  the 
same  as  the  ratio  of  the  corresponding  two  on  the  right.     Thus 

24  :  8  :  6=  12  :  4  :  3; 
although      24 -f-  8  -f- 6  =  £     and     12-^4^3  =  1. 

Some  authors  define  :  as  the  precise  equivalent  of  ~  ;  but 
even  they  generally  make  some  distinction  in  the  use  of  the 
two  signs. 

V.    Illustrations. 

8l.      —5—4—3—2—1        o        1        2        3        4        5        6 

j ! 1 1 ! ! ! ! ! 1 ! [ 

Suppose  that  from  the  point  marked  o  on  this  line  I  meas- 
ure off  equal  distances  to  the  right  and  left.  These  distances 
I  call  unit  distances  or  steps,  and  the  point  3  is  three  steps  to 
the  right  of  the  origin  of  measurements,  while  the  point  —  4  is 
four  steps  to  the  left  of  that  origin. 

Then  the  statements 

—  10+7  +  4  —  8  =  7  — 8  +  4+  10  =  0+11  —  18  =  —  7 

may  be  translated  :  "  If,  starting  from  a  point  10  steps  to  the 
left  of  the  origin,  I  go  7  steps  right,  then  4  right,  then  8  left,  I 
get  to  the  same  place  as  if,  starting  from  7  steps  to  the  right  of 
the  origin,  I  go  8  steps  left,  then  4  right,  then  10  left,  or  just 
the  same  as  if,  starting  from  the  origin,  I  go  1 1  steps  right  and 


46         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

then  1 8  left;  in  fact,  by  each  of  these  routes  I  find  myself  7 
steps  to  the  left  of  the  origin." 

Again,  suppose  I  pay  out  $10,  take  in  $7,  take  in  $4  more, 
and  then  pay  out  $8.  It  will  be  the  same,  upon  the  whole,  as 
if  I  had  merely  paid  out  $7. 

Still  again,  let  A,  B,  C,  D  stand  for  four  events.  If  I  know 
that  A  happened  10  years  ago,  B  7  years  after  A,  that  C  will 
happen  4  years  after  B,  and  finally  that  D  happened  8  years 
before  the  time  when  C  will  happen  ;  then  D  happened  7  years 
ago. 

Let  the  student  give  other  illustrations. 

Imagine  that  after  going  2  steps  right  and  then  7  left,  I 
find  myself  10  steps  to  the  left  of  the  origin.     Where  was  I  ? 

I  retrace  my  steps,  going  from  10  left,  7  steps  right,  and 
then  2  left,  and  I  find  myself  5  steps  left  from  the  origin.  Or 
again,  I  notice  that  I  have  come  upon  the  whole  5  steps  left 
from  the  starting  point,  and  retracing  these,  I  get  as  before  a 
point  5  steps  left  from  the  origin. 

In  symbols : 

-  10  -  (+  2  -  7)  =  -  10  -  2  +  7  =  -  5, 
and      -  10  -  (+  2  -  7)  =  -  10  -  (-  5)  =  -  10  +  5  =  -  5 

an  illustration  of  association  and  sign-reversal. 

Of  the  latter,  here  are  others:  less  of  westing  is  more  of 
easting ;  less  of  spent  is  more  of  saved  ;  to  lighten  one's  bur- 
dens is  to  add  to  one's  strength  ;  taking  away  cold  is  making 
warm. 

But  there  are  problems  where  negative  numbers  are  non- 
sense. A  man  cannot  live  a  negative  number  of  years.  A 
pond  cannot  be  —  4  feet  deep.  A  table  cannot  have  —  3  legs. 
No  one  is  —  6  feet  tall. 

82.  If  I  twice  repeat  three  steps  to  the  right  from  the 
origin,  I  go  6  steps  to  the  right.  If  I  twice  repeat  3  steps  to 
the  left,  I  go  6  steps  to  the  left.  If  I  twice  retrace  3  steps  to 
the  right,  I  go  6  steps  left.  If  I  twice  retrace  3  steps  to  the 
left,  I  go  6  steps  right. 


ILLUSTRA  TIONS. 


47 


6;  3X— 2  =  — 6;(-3)X— 2=6. 

Distances  to  the  right  on  the  lever 


In  symbols: 

3  X  2  ±=  6  ;  (-  3)  X  2  =  - 

Again,  I  have  a  lever. 
are  positive ;  to  the  left,  negative.  A  pull  up  on  the  lever  is 
positive ;  a  pull  down,  negative.  The  unit  pull  is  that  of  one 
pound  one  inch  from  the  fulcrum,  and  is  positive  if  it  tends  to 
lift  the  right-hand  end  of  the  lever. 


} 


9- 


* 


* 


Plainly,  a  pull  up  of  3  lbs.,  2  in.  to  the  right  of  the  fulcrum 
is  a  6-unit  positive  pull ;  a  pull  down  of  3  lbs.,  2  in.  to  the  left 
of  the  fulcrum,  is  a  6-unit  negative  pull ;  a  pull  up  3  lbs.,  2  in. 
to  the  left  of  the  fulcrum,  is  also  a  6-unit  negative  pull ;  and 
finally,  a  pull  down  of  3  lbs.,  2  in.  to  the  left  of  the  fulcrum,  is 
a  6-unit  positive  pull. 

83.  Suppose  I  am  between  a  point  2  steps  to  the  right  of 
the  origin  and  one  3  steps  to  the  right  of  the  origin.  I  can 
indicate  my  position  by  saying  what  part  of  the  way  I  am  from 
2  to  3  ;  and  of  course  I  can  indicate  my  position  by  a  fraction. 
Thus,  the  point  f  is  half  way  from  2  to  3.  Such  conventions 
serve  to  introduce  fractions  into  all  the  illustrations  that  we 
have  given.  Notice,  however,  that  just  as  negative  numbers  are 
sometimes  non-sense,  so  are  fractions.  A  polygon  cannot  have 
a  fractional  number  of  sides.  A  ball  cannot  be  thrown  2\  times. 
A  surface  has  2  dimensions,  a  solid  3  ;  there  is  nothing  between. 

84.  Not  only  are  there  distances  to  be  expressed  by  frac- 
tions, but  also  distances  which  must  be  expressed  by  incom- 
mensurables.     For  example,  the  diagonal  of  a  square  whose 


48         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

side  was  the  unit  distance  would  be  4/2.  In  fact,  if  we  draw 
two  lines  at  random  we  cannot  take  for  granted  that  they  have 
any  common  measure.  Plainly,  it  is  infinitely  improbable  that 
one  should  exactly  measure  the  other :  nor  is  it  a  whit  more 
probable  that  a  half,  a  third,  a  quarter,  that  any  named  fraction 
of  the  one  should  exactly  measure  the  other. 

If  on  the  line  of  §  82  a  point  moves  from  2  to  3,  it  passes 
through  positions  at  all  distances  from  the  origin  both  frac- 
tional and  incommensurable  between  2  and  3.  The  same  is 
true  of  distances  from  the  fulcrum  on  our  lever.  Likewise,  if 
in  the  latter  problem  the  weight  of  pull  is  continuously  in- 
creased, as  it  would  be,  very  nearly,  if  due  to  water  gradually 
flowing  into  a  containing  vessel ;  then,  the  weight  of  pull  takes 
both  fractional  and  incommensurable  values.  Thus  the  num- 
ber expressing  the  pull  in  terms  of  the  unit  pull  might  be  the 
product  of  two  incommensurables.  But  as  the  pull  varies 
directly  with  its  weight  and  its  distance  from  the  fulcrum,  these 
products  must  lie  between  the  products  of  commensurables 
and  be  hemmed  in  by  them  as  closely  as  one  pleases. 

VI.    Growth  and  Rate. 

85.  We  have  now  introduced  the  main  part  of  the  notation, 
the  fundamental  conceptions,  and  the  material  of  elementary 
algebra.  There  remain  for  discussion  the  expression-numbers 
arising  from  the  attempt  to  take  even  roots  of  negatives. 

Before  entering  upon  that  discussion,  however,  we  shall  ex- 
hibit some  of  the  operations  and  their  results  in  a  new  light. 

86.  Suppose  x  to  be,  in  succession,  all  numbers  from  zero 
to  a-number-as-large-as-you-please.  We  say  x  grows  positively 
from  zero  (o)  to  infinity  (-f-  00  ).  On  the  other  hand,  if  x  is,  in 
succession,  all  numbers  from  o  to  —  00 ,  we  say  that  it  grows 
negatively. 

Any  positive  number  a  is  the  result  of  a  positive  growth, 
say,  is  a  positive  growth;  while  —  a  is  the  result  of  a  negative 
growth.     The  sum  of  two  growths  is  the  result  of  the  growth 


GROWTH  AND  RATE.  49 

from  either  that  would  have  given  the  other  from  zero.     We 
easily  extend  this  to  any  number  of  growths.     Thus, 

OA  +  AB  +  BC  +  CD  +  DE  +  EF+FG  +  GH '  =  OH. 

G    F  B  oA  C  E  D  H 

Evidently  this  equation  would  still  be  true  if  we  were  to 
change  the  position  of  the  letters  upon  the  line  in  any  imagi- 
nable way.  Quite  independent  is  it  too  of  what  we  choose  for 
a  unit  growth,  and  of  whether  any  or  all  of  the  growths  added 
are  incommensurable  with  that  unit  growth. 

Subtraction  is  included  in  addition  and  need  not  delay  us. 

87.  Let  y  grow  from  zero  so  that  always  y  =  ax,  where  a  is 
a  positive  number  that  does  not  grow.  If  x'  andy  are  two 
corresponding  fixed  values  of  x  and  y,  then  always 


y 
y  —  y'  =  a(x  —  x'),    and     a  ==  — 


We  say  that  y  grows  with  x  at  a  uniform  rate  a. 

Did  we  have  y  =  ax  -\-  b,  where  b  is  another  constant,  we 

y  —  y' 
should,  as  before,  have  — — — -,  =  a.     The  rate  of  growth  of  y 

compared  to  x  is  the  same  as  before  ;  but  y  grows  from  b,  while 
x  grows  from  zero.  In  other  words,  y  has  the  start  b  of  x,  and 
keeps  that  start. 

Did  we  have  y  =  —  ax  or  y  =  —  ax  ±  b,  we  should  say  that 
y  grew  against  x  at  the  uniform  rate  a. 

When  y  =  ax,  we  say  that  y'  is  the  result  of  ys  growing 
from  zero  at  the  uniform  rate  a,  while  x  grows  from  zero  to  x' . 

88.  Suppose  now  y  =  x2y  and  that  as  before  x'  and  y'  are 
corresponding  fixed  values  of  x  and  y.  As  x  grows  from 
—  co  to  zero,  y  grows  against  x  from  -\-  00  to  zero ;  and  as  x 
grows  on  from  zero  to  -f-  00  ,  y  grows  with  x  from  zero  to  -f-  00. 
What  is  the  rate  of  growth  ? 


50         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

y  —  y' 

Consider  the  fraction  - —. .     We  have 

x  —  x 

X  —  X  X  —  x  ■ 

Evidently  it  changes  its  value  as  x  changes  its  value  ;  is  less 
than  2x'  when  x  is  less  than  x\  and  is  greater  than  2x'  when  x 
is  greater  than  x' ;  while  at  the  moment  when  the  growth  of 

y — y  y  —  y 

x  reaches  x'  and -.  becomes  —. -. ,  its   value   is   2xf. 

x  —  x  x  —  x 

y  —  y 

We  call  the  expression  —7- 7  the  rate-fraction  and  say  that  it 

gives  the  rate  of  growth  y  compared  to  x  when  x  =  x' . 

y  —  y    o 

On  the  face  of  it,  — 7  =  -,  and  might  be  anything  you 

X         X  o 

o 
please:  for  k  X  o  =  o  and  so  k  =  — ,  no  matter  what   number 

y — y . 

k  may  be.     In  our  present  problem  — -, 7  is  2x'  because  of 

x        x 

y  —  y' 

the  law  connecting  the  growths  of  x  and  y  as 7  becomes 

x       x 

y  —  y  y  —  y 

or  grows  to  be  —, 7 .     In  fact  —, 7  is  hemmed  in  as  close 

fe  X    —  X  X    —  X 

y  —  y'  ' 

as  you  please  by  values  of -, . 

X  X 

Thus,  if  x'  =  o  and  we  imagine  x  to  take  the  values  ±  1, 

11  1  1  1 

fc  — ,  + ,  ± 1  ±  .  ±  — ir  1  where  n  is  as  large 

10 '        100 '        1000'        1000000'        I0W  b 

jf — y 

as  you  please,  is  infinity,  the  corresponding  values   of 7 

X  —  X 

ft  III  I  I 

are   likewise   ±  !>  ±  —  1  ± >  db »  ± >  x  — «  > 

'       10         100         1000         1000000         10* 

1/  —  _j/r 

so  that  as  x  nears  4T  or  zero  from  either  side  — 7  also  nears 

x  —  x 

y — y 

zero.     But  at  the  same  time  it  nears  — -, ,  and  so  tells  us 


GROWTH  AND  RATE.  5  I 

y  — y 

that  —, 7  is  not   "anything  you   please;"  but,  on  the  con- 

x  —  x 

trary,  is  the  definite  number  zero. 

y  —  y       •  y — y 

Drop  the  accents  from  — -, ;  and   it  becomes ,   a 

r  X    —  X  X  —  X 

variable,  the  varying  rate  of  growth  of  y  compared  to  x\  and 
does  itself  grow  with  x  at  the  constant  rate  2.  At  first  when 
;tris  negative  oo  the  rate  of  growth  of  y  compared  to^ris  against 
x  at  the  rate  2  X  oo .  (We  mean  by  this  that  however  large  in 
absolute  value  the  number  may  be  by  which  x  is  expressed, 
that  by  which  the  rate  of  growth  is  expressed  will  be  twice  as 
large.)  As  x  increases,  becoming  less  negative,  the  rate  like- 
wise increases  and  twice  as  rapidly  as  jit,  becoming  less  and  less 
against  x ;  until,  when  x  is  zero,  the  rate  is  zero.  At  this 
instant,  the  rate  changes  from  being  against  x  to  being  with  x, 
and  from  now  on,  increasing  as  before,  twice  as  fast  as  x  does, 
becomes  2  X  oo  ,  when  x  becomes  oo. 

At  the  beginning  of  this  section  we  said  that  as  x  grew 
from  —  oo  to  zero,  y  grew  from  -{-  oo  to  zero.  It  would  have 
been  more  accurate  to  have  said  that  y  grew  from  ( —  oo)2  =  oo2 
to  o ;  that  is,  that  y  grew  from  a  number  as  many  times  greater 
than  the  opposite  of  x  as  the  opposite  of  x  was  times  greater 
than  unity,  no  matter  how  large  the  opposite  of  x  might  be. 

89.  Whatever  the  law  may  be  by  which  the  growth  of  y  is 
connected  with  the  growth  of  x,  we  define  the  rate  of  growth 

y  —  y 

of  y  compared  to  x  by  this  same  fraction  - — -,  and  the  rate 

y'  —  V 

of  growth  when  x  =  x'  by  the  fixed  fraction  ~ — —. . 

x  —  x 

Let  the  student  determine  the  rates  for — 

y  =  2x  —  3,  when  x'  =  -  3,  -  2,  —  1,  o,  1,  2,  3; 

f-  =  2X,  "      4  =  r~  7,  o,  +  $  ; 

y  =  $x2  —  5,  .  "      x'  =  —  11,  —  1000,  +20; 

y  =  x3,  "      x'  =  2  ; 

j3  =  3^4  4-  2jr3-f-^r2  +  I,      "      x'  =s  o,  I. 


52         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

Also,  in  each  of  the  above,  what  is  the  rate  of  ;r-growth 
compared  to  ^-growth  ?     In  other  words,  what  is  the  value  of 

x  —  x  ~ 

y—y 

If  y  =  2x  -f-  5  and  z  =  y2,  what  is  the  rate  of  ^-growth 
compared  to  jr-growth,  when  x'  =  3  ? 

90.  Suppose  tf  >  1  and  ^  ==-  a? ;  what  is ? 

When  ^  involves  an  even  root,  is,  say,  f ,  y  might  be  nega- 
tive :  these  values  of  y  we  rule  out.  We  are  considering 
growths,  and  y  does  not  grow  from  one  value  to  another  unless 
it  takes  in  succession  all  values  between  the  two.  Though 
the  values  are  as  close  together  as  you  please  and  y  takes 
hundreds  of  thou  sands  of  millions  of  values  between  thm,  it 
does  not  strictly  grow  from  one  to  the  other.  Thus,  above,  y 
cannot  grow  from  —  a1  to  —  ah ;  for  though  it  may  have  as 
many  values  as  you  please  between  these  two,  there  are  as 
many  other  intermediate  values,  for  example  —  a,  which  y 
cannot  have.  On  the  other  hand,'  y  does  grow  from  ah  to 
ah  when  x  grows  from  \  to  f,  for  y  then  takes  all  values  between 
ah  and  ah. 

Then,  as  x  grows  from  —  00  through  o  to  +  00,  y  grows  from 
o  through  1  to  -f-  00. 

y'  —  y'       1  —  1  ,       .     ,     .  ah  —  1 

At  x'  =  o,  —, --,  = and  the  inclusives  are   — : — 

.r  —  .r       o  —  o  h 

a~h 1 

and -j — ,  where  h  is  a  positive  number  growing  zero-ward, 

so  that  h  and  —  h  are  inclusives  of  zero. 

ah 1  a~h  —  1 

To  prove  that  — 7 —  and  -. —  really  are  inclusives  of 

y1  —  y'  y  —  y' 

~ ~  for  xf  =  o;  we  notice,  first,  that  they  are  what , 

x   —  X  x        * 

becomes  when  we  put  in  turn  x  —  h  and  x—  —h  whiles' remains 

ah 1 

zero.     We  then  show  that  there  is  no  least  value  of  — 7 —  ,  no 

,  a~h  —  1         ,   ,         ,  ah  —  1       a~h  —  1 

greatest  value  of  — — r     ,and  that  always  — ■; —  >  — — j — . 


GROWTH  AND   RATE.  53 


ar  —  I 
There  is  no  least  value  of  — j —  if,  for  o  <  k  <  //, 

ak  —  I       ah  —  I 


<  — : —  ;  i.e.,  \io<h  —  k  —  hak  +  ^A. 


£       -      h 

When  k  =  I  and  ^  =  2  the  condition  becomes o <i  —  2a -\- a2, 
which  is  o  <  (i  —  of,  and  true  because  all  squares  are  positive. 
Again,  when  k  =  2  and  h  =  3,  the  condition  becomes 
o  <  1  —  3a2  +  2a3,  or  o<a(i  —  a)2  -f  (a2  —  i)(a—i);  and  this 
is  true  because  a(i  —  a)2  and  (a2  +1)  (a  —  1)  are  separately 
greater  than  zero.  In  like  manner  we  could  prove  the  condi- 
tion to  be  fulfilled  when  k  =  3  and  h  —  4.  It  is  better,  how- 
ever, to  show  that  if  true  for  k  and  h  any  two  consecutive  num- 
bers, it  remains  true  when  we  increase  both  k  and  h  by  unity : 

that  if  O  <  I  —  \k  +  \)ak  +  kak+\ 

then  also  O  <  I  -  {k+  2)ak+1  +  {k  +  i)ak+2. 

This  is  easy  enough,  for  the  last  expression  is  the  sum  of 
a  times  1  —  {k  -f-  i)#*  -j-  &a?+l,  positive  by  the  first  inequality, 
and  (ak+1  —  1)  (a  —  1),  the  product  of  two  positives. 

Therefore  the  first  inequality  does  involve  the  second,  and 
the  condition  holding  for  k  =  1  and  h  —  2  holds  for  k  =  2, 
h  —  3,  for  k  =  3,  h  —  4,  for  £  =  4,  £  =  5,  .  .  .  ,  for  k  —  any 
integer  and  h  =  k-\- 1.  Still  more  does  it  hold  if  ^  and  //  are 
integral,  and  h>  k  -\-  1. 

/  r 

Suppose  now  ^  and  ^  are  fractions  -  and  -,  where  r  >p>o. 

i  r- 

The  condition  becomes  o  <  r  —  p  —  raq  -\-  paq,  precisely  what 

1 
we  had  before  with  r,  /,  and  cfl  in  place  of  h,  k,  and  a ;  and  is 
true  because  these  numbers  fulfil  all  the  conditions  imposed 
upon  hy  k,  and  a. 

ak  —  I       ah  —  1 
•.  — 7 —  <  — -j —  for  o  <  k  <  h 


54         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

when  k  and  h  are  commensurable.  But,  since  the  values  of 
the  expression  when  k  and  h  are  incommensurable  are  hemmed 
in  by  the  values  when  k  and  h  are   commensurable,  the  in- 

equality  is  always  true,  and  there  is  no  least  value  of 


Neither  is  there  a  greatest  value  of 


k 

a~h  —  i 


This  will  be  found  to  hang  upon  the  inequality 

o<h—k— ha~k-{-ka-h  or  o<h— k— /;(-)  +  &[-)    for  o<k<h. 

The  demonstration  will  proceed  precisely  as  before,  except 
that   where   we    had   (ak+1  —   i){a  —  i),   we   shall    now   get 

[~k+i  —  Ol-  —  l)y  tne  product  of  two  negatives  instead  of  the 

product  of  two  positives. 

ah i       a~k  —  I 

Finally,  — -, —  > t —    for  h  >  o  and  k  >  o,   whether 

h  >  k,  h  =  k,  or  h  <  k. 
If  h  =  k, 


T„  ,        ,       <f  —  I       «*  —  I       a~k  —  I 


Tr  ,        .       #*  —  i       a-*  —  i      a~*  —  I 


Thus,    as   stated,    — ^ —    and   j —   are   inclusives    of 


ah  —  I         .    a~h  —  I 
1 

j/'  —  j/' 

-7 7  for  x  =  o  and  y  =  a*.- 

x   —  x  * 

91.  To  fix  our  ideas,  suppose  #  is  2,  so  that  j  as  2* ;  then 

y  —  v  2;*  —  1      2~h  —  1 

— ; 7  for  x'  =  o  lies  between  -r —  and -, — ;  i.e.,  be- 

x  —  x  h  —  h 

tween    n{  \/2  —  1)   and  n{  \/2  —  1)  -r-  y/2,  where  n  =  j.      For 


GROWTH  AND  RATE. 


55 


convenience  in  calculation,  put  n  =s  I,  2,  4,  8,  .  .  .  We  shall 
find  the  rate  fraction,  call  it  r,  hemmed  in  as  follows : 


n  =  1,  1.  >  r  >  0.5; 

n  =  2,  0.83  >  r  >  0.58 

«  =  4,  0.76  >  r  >  0.64 

n  =  8,  0.73  >  r  >  0.66 

#  =  16,  0.71  >  r  >  0.67 


n  —  4096,         0.693 1  >  r     >  0.6929 ; 


and  the  rate  to  the  nearest  thousandth  is  0.693. 

92.  To  find  the  rate  when  x'  is  different  from  zero  we  have 


y-y 


X    —  X 


But  when  ^  >  o, 


Mr'  +  A 


2-*'        2*  —  2*         2' 

—  >  r^ — ~zr  >  ~ 


h_2x> 


since  this  may  be  written, 


,    2h  —    I  ,1—1  ,    2~h  —  I 

2*  . j-      >  2X  . >  2X\ r— 

h  O—O  2_A 


Therefore    the    ratio     at    x'  =  2x'r  —  y'  X  0.693.       When 

,  .      .        *.                    0.603  0.603 

x  —  —  2,  —  1,  o,  1,  2,  this  gives  for  the  rate  — -- ,  — —,  0.693, 

4  2 
2  X  0.693,  4  X  0.693. 

y  —  y 

The  varying   rate  of   growth  — — —    has  always   the  ratio 

0.693  to  the  varying  y  ;  i.e.,  grows  with  y  at  the  uniform  rate 

0.693.     We  say  that  y  grows  logarithmically  with  regard  to  x, 

s  rate  of  growth 

the  log2  y\  and  the  number  0.093,  = ^ — -^ — ,  we  call 

&2-^  ^        growing  number 


$6         AN  INTRODUCTION   TO    THE   LOGIC  OF  ALGEBRA. 

the  logarithmic  growth  rate.  Thus,  27  is  the  result  of  allowing 
y  to  grow  from  unity  at  the  logarithmic  rate  0.693  with  regard 
to  x  growing  from  zero  to  7.  Or,  dropping  the  y  and  x,  we 
say  that  27  is  the  result  of  unity's  growth  at  the  logarithmic 
rate  0.693  with  regard  to  zero  growing  to  7. 

93.  But  unity  is  also.  (27)0  and  27  =  (27)1.  So  27  is  likewise 
the  result  of  unity's  growth  at  another  logarithmic  rate  with 
regard  to  zero  growing  to  1 .    What  is  this  new  logarithmic  rate  ? 

(27f  —   I 

Why,  — -7 when  h  becomes  zero.    We  may  otherwise  write 

27h  —  I 

it  7  . ; —  ;  and  as  here  the  last  factor  is  0.693,  the  new  rate 

7/1 

is  7  X  0.693. 

Equally  well  can  we  get  27  by  unity's  growth  at  the  loga- 
rithmic rate  3  X  0.693  with  regard  to  zero  growing  to  f ,  or  by 
unity's  growth  at  the  logarithmic  rate  k  X  0.693  with  regard  to 

7 
zero  growing  to  t*. 

Call  k  X  0.693,  r,  and  in  place  of  7  write  b.     We  see  at 

once  that  any  number  2b  can  be  reached  by  unity's  growth  at 

any  assigned  logarithmic  rate  r  with  regard  to  zero  growing  to 

b 

-  X  0.693. 

Fixing  the  logarithmic  rate  r  fixes  the  base  by  whose  power- 
ing unity  grows.     The  base  is  in  fact  2?  *  °-693. 

Of  great  importance  is  the  base  for  unit  logarithmic  rate, 
2i  +0.693  _  2m*-..,  which  we  call  the  natural,  hyperbolic,  or 
Naperian  base,  and  denote  by  e.  At  once,  because  f  >  1.46  >  f , 
23  >  e  >  23,  or  2.8  >  e  >  2.5.  More  accurately,  e  is  the  incom- 
mensurable 2.7 1 828 1 8.  ...  It  has  been  defined  as  the  base  by 
whose  powering  unity  grows  at  the  logarithmic  rate  unity;  it  is 
also  the  result  of  letting  unity  grow  at  the  logarithmic  rate  unity 
with  regard  to  zero  growing  to  one.     Compare  §  87. 

Prove  that  by  the  powering  of  any  base  a,  unity  grows  at 
the  logarithmic  rate  log,,  a ;  and  also  that  a  is  the  result  of 
unity  growing  at  the  logarithmic  rate  unity  with  regard  to  zero 
growing  to  loge  a. 


GROWTH  AND  RATE.  57 

The  natural  logarithms,  that  is,  the  logarithms  to  the  base 
e  of  the  numbers 

I,         2,  3,         4,  5,  6,         7,  8,         9, 

are 
o.ooo,   0.693,    1.099,    1*386,    1.609,    1.792,    1.946,   2.079,  2.197. 

Notice  that  1.386  =  2  X  0.693  and  1.792  =  0.693  -f-  1.099. 
Why  these  relations  ? 

At  what  logarithmic  rate  does  unity  grow  by  the  powering 
of  12,  15,  27,  2.5,  3i,  i? 

What  must  be  the  growth  of  zero  for  the  above  bases  that 
unity  may  grow  to  20?  to  10?  to  —  5  ? 

94.  The  conception  at  the  end  of  §  97  may  be  used  to 
approximate  to  the  number  e. 

When  x  grows  from  o  to  — ,  e*  grows  from  1  to  e* .     Had  ^ 

kept  the  rate  of  growth  it  had  for  x  =  o,  i.e.  grown  uniformly 
instead  of  logarithmically  with  regard  to  x,  it  would  have  grown 

to   1  -| — .      As  it  has   really  grown   ever  faster   and    faster, 

1  -I  1 

en  >  1  A — .     On  the  other  hand,  e  n  <  1 . 

1   n  n 


■■■  t-J  >  -  >  HH  - 


whatever  positive  number  n  may  be.     We  may  otherwise  write 
the  e  limits 


fn+i\n  +  1  fn+i\H 


Here  the  superior  limit  is  just  -th  larger  than  the  inferior, 

and  we  therefore  get  e  to  within  less  than  an  rcth  part  of  itself. 
For  instance,  if  n  =  1000000,  we  know  that  e  does  not  differ, 
since  e  is  less  than  3,  3  units  in  the  sixth  decimal  place  from 
1. 00000 11000000. 


58         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

By  exact  parity  of  reasoning, 

E.g.,  1.0000031000003  >  e3  >  1. 000003 100000°. 

Because  e  <  3,  we  have 
e3  <  27,    1.000003  e3  <  27.000081,    1.0000032  e3  <  27.000162, 
and  1.0000033  e3  <  27.000243. 

Consequently  1. 000003 10000°3  exceeds  1. 000003 I0OO0O°  by  less 
than  0.000243,  and  either  limit  comes  still  closer  to  e3.  By  tak- 
ing n  large  enough  we  can  of  course  get  e3  to  any  desired  de- 
gree of  accuracy. 

Now   notice.       Not    only   is   e*   as    nearly   as   you    please 

f  I  -J — J  ,  when  n  is  taken  large  enough,  but  also  e*  =  ( 1  -| — J    • 


-)■ 


•    .-.  when  n  is  large  (1  +  -j     =  (1  +  - 
This  result  might  have  been  foreseen,  for 

[.+r-(>+=r.-(«+3i. 

since  the  x  multiplying  the  two  n's  in  the  middle  expression  is 
meaningless,  n  being  merely  any  sufficiently  large  number. 

A  continuation  of  this  reasoning  shows  that,  to  any  desired 
degree  of  approximation, 

<-(-r=(-r.*     ■ 

„a  ,-.=(,-ir=('-f=('+ir=(,+f"- 


GRAPHS.  59 

In  brief,  e*  =  ( i  +  -J     =  ^i  +  -J    holds  for  x  and  for  n 

either  positive  or  negative,  provided  n  is  large  in  absolute 
value. 

95.  Rough  examples  of  uniform  and  logarithmic  growth  are 
furnished  by  money  put  out  at  simple  and  at  compound  in- 
terest respectively.  Did  the  interest  come  in  not  merely  yearly 
or  half-yearly,  or  even  every  day  or  minute  or  second,  but  all 
the  time,  the  examples  would  be  perfect.  What  we  call  the 
rate  of  interest  corresponds,  in  the  one  case,  to  what  we  have 
called  rate  of  uniform  growth ;  in  the  other,  to  what  we  have 
called  rate  of  logarithmic  growth.  In  the  one  case,  the  money 
grows  by  equal  amounts  in  equal  times ;  in  the  other,  it  grows 
by  equal  multiples  of  itself  in  equal  times,  is  equally  multiplied 
in  equal  times. 

The  more  often  we  compound,  the  yearly  rate  remaining 
unchanged,  the  greater  will  be  the  amount  of  a  given  sum  of 
money  put  out  for  a  fixed  time.  Show,  however,  that,  no 
matter  how  often  the  compounding,  $100  at  io#  per  annum 
for  10  years  could  not  amount  to  so  much  as  $271.83. 


VII.    Graphs. 

96.  We  can  picture  to  the  eye  some  of  the  results  of  the 
preceding  sections. 

We  represented  positive  and  negative  numbers  by  distances 
measured  along  a  line  to  the  right  and  left  of  a  fixed  point 
called  the  origin.  Equally  well  are  they  represented  by  dis- 
tances from  the  line  upward  and  downward. 

In  the  equation  y  —  \x,  represent  x  by  a  distance  along  the 
line,  and  y  by  a  distance  measured  at  once  from  the  line  and 
from  the  end  of  the  distance  x. 

Thus  below,  if  x  is  4,  y  is  the  distance  2  of  the  point  (4,  2) 
from  4,  on  the  line  of  x's. 

Equally  well,  of  course,  we  can  write  the  equation  x  =  2yy 
and  say  that  if  y  =  2,  x  is  the  distance  4  of  (4,  2)  from  the 
point  2  on  the  line  of  y's. 


60  AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

When  x  takes  the  series  of  values 

—  6,-5,-4,-3,-2,-1,  o,  i,  2,  3,  4,  5,  6, 

y  is   -  3,  -  |,  -  2,  -  |,  -  i,  -  i,  o,  J,  i,  |,  2,  |,  3, 

and  is  represented  by  the  distances  of  the  dots  on  the  broken 
line  from  the  line  of  x's. 


»9        -6 


-6       -5 


-2       -1 


■^3 


fi&**u& 


Line  ofz%  or  y  —  o. 


By  geometry,  these  dots  are  in  one  straight  line,  and  the 
dots  gotten  by  taking  any  value  whatsoever  of  x  would  also 
lie  on  this  line.  Further,  all  points  on  the  broken  line  are 
points  (x,  y),  i.e.  points  such  that  the  y  of  any  one  of  them  is 
half  the  x. 

We  call  the  line  the  graph  of  y  =  \x. 

In  y  =  ax,  a  is  simply  y  for  x  =  I. 

Let  the  student  construct  the  graphs  of  y  =  $x,  x  =  —  2%yf 
y  =  2*  -  3,  y  =  x  |/2  +  I,  2\X  =  -  $\y. 

Having  given  a  number  of  distances,  show  how  by  a  graph 
to  get  a  given  multiple  of  all  of  them. 

To  construct  the  graph  of  y  —  2X,  we  have  the  points  (x,  y) 
as  follows : 


(-4,  A).  (-3,  i),  (-2,  i),  (- 1,  i),  (o,  i),  (i,  2),  (2, 4),  (3,  8),  (4(  16). 


GRAPHS. 


6l 


The  graph  is  sketched  below.      It   is   also  the   graph  of 
x  =  log2  y. 


8 

/       / 
/      / 

4 

U 

rfi 

i  / 

2 

1/ 

1/ 

l 

-4       -3         -2          -1 

0         12           3 

•4 

The  graphs  of  all  equations  y  =  ax  are  of  this  sort,  if  a  >  I. 
They  all  pass  through  (o,  i)  on  the  line  of  ys,  all  grow  steeper 
for  increasing  x,  and  for  any  x  the  graph  that  has  the  greatest 
a  is  steepest. 

Construct  the  graphs  of  y  =  \/2x,y  =  1. 1*,  y  =  i.oi*,  y  =  I*, 
^  =  (i)*i  y  =  2**,  log§  *=*J*  log5  ^  =  *• 

y  —  y' 
97.   In  the  graph  of  y  =  ax,  al  = 7  is  called  the  slope 

°f  the  graph,  which   is  in   this  case   a  straight   line.     In  the 


graphs  of  y  =  x2,  y  =  ax,  and  in  other  curved  graphs, 


y  -y 


is  the  slope  of  the  curves  at  (x\  y'). 

What  is  the  slope  of  y  =  2x  -f-  3  ?  the  slope  of  y  =  x2  at 
(o,  o)  and  where  X?  =  2,  3,  —  I  ?  of  y  —  2X  at  (o,  1)  and  where 
**  =  2,  5,  —  3  ? 

In  general,  the  slope  of  a  line  joining  arbitrary  points  (x,  y), 

v  —  y' 
(xr,  yf),  is 7.     If  we  put  the  points  on  a  curved  graph,  the 


62 


AN  INTRODUCTION   TO    THE   LOGIC  OF  ALGEBRA. 


line  cuts  across  the  graph.     If  the  points  run  together,  (y  =  y\ 


jc  =  x'\  but  remain  on  the  graph, 


y  -y 


7  is  the  slope  of  a 


line  touching  the  graph  at  (xf,  y'). 

Thus  (compare  §  93),  1,  0.83,  0.76,  0.73,  0.71  are  respectively 
the  slopes  of  lines  cutting  across  the  graph  of  y  =  2X  from  the 
points  (o,  1)  to  the  points  (1,  2),  (f,  |/2),  &  J/2),  (i  f/2),  (T\,  lfo) ; 
while  0.693  is  the  slope  of  a  line  touching  the  graph  at  (o,  1). 
Our  approximating  to  the  value  of  the  rate  fraction  is  thus 
finding  the  slope  of  a  cutting  line  as  the  points  of  cutting  run 
together,  making  it  a  touching  line. 


98.   Of   course,   when  y  =  ax   and   x  —  x' 


k 


y  —  y 

X  -  x' 


is 


Suppose,  as  shown  below,  that  we  give  x'  the  series 


of  values 

—  2/1,     —  h,    o,    //,    2h,     $k,    4/1,     5/1,     6k,    yh. 
And  further,  let  the  line  joining  (x,  y)  with  (V,  y')  cut  the 


1J 
/ 

3 

//   1 

2 

1 

/      / 

-3ft. 


-2ft 


■Ui 


Oft, 


V, 


line  of  ;r's  where  ■#•=:/£.  By  geometry,  ^'  —  k,  y' ,  and  the  cut- 
ting line  form  a  triangle  similar  to  that  formed  by  x  —  k,  y, 
and  the  cutting  line.  They  might  be  the  triangles  h,  p'y  3/1,  and 
Jt,  p,  4k. 


We  have 


x  —  k       y 
x  —  k      y 


.\k  = 


an  —  1 


=  X 


GRAPHS.  63 

and  x>-k  =  <?—  , 

no  matter  what  the  value  of  x' . 

h 

In  the  figure  we  have  supposed  — to  be  2//,  and  start- 

&  rr  ah  —  1 

ing  with  the  point  (o,  1)  have  found  a  number  of  points  upon 

the  graph. 

Show  that  the  line  touching  the  graph  of  y  =  2*  at  (o,  1) 
cuts  the  line  of  x's  at  (—  1.46,  o). 

Prove  that  a  line  touching  x  =  logay  at  (x',yf)  cuts  the  line 

of  xs  at  \x'  —  -. ,  o),  i.e.  at  {x'  —  loga  e,  o). 

The  point  (4,  3)  is  on  y  =  a? ;  what  is  the  value  of  a  ? 

In  the  figure  just  considered,  suppose  h  =  2 ;  then  the 
graph  of  y  =  |/V*  is  touched  at  (o,  1)  by  the  line  joining 
(—  2/1,  o)  with  (o,  1).  Of  course,  all  points  of  the  graph,  save 
(o,  1),  lie  wholly  above  that  line;  but  the  point  (1,  tye)  is  only 
a  little  above,  the  point  (|,  fye)  still  less  above,  and  so  on. 
Furthermore,  were  we  to  go  out  on  the  touching  line  till  just 
under  the  point  (£,  \/e),  and  then  start  the  construction,  join- 
ing the  point  just  reached  with  (—  3 J,  o),  the  joining  line  would 
lie  closer  to  (1,  \/e)  than  our  former  line.  By  starting  the  con- 
struction closer  and  closer  to  o,  1,  we  should  get  lines  and 
points  closer  and  closer  to  the  graph  of  y  =  \/e,  and  as  close  as 
one  pleased,  when  at  a  finite  distance  from  the  line  of  y's. 

This  is  a  geometrical  interpretation  of^  =  li+-)' 


Part  Second. 

DOUBLE    NUMBERS 


I.    Integral  Double   Numbers  and   the  Simpler 
Operations. 

99.  We  now  consider  the  even  roots  of  negatives  referred 
to  in  §  87. 

Assuming  the  law  am  X  bm  =  (a  X  b)m,  we.  have 


V—  4  =  V 4  X  -  1  =  V-  r  X  4  =  V4  •  V-  I  =  V-  l  •  V4 
=  ±  2.  |/—  I  =  ±  |/—  1 .   2. 
Similarly, 

^-9=±3  V-I>V-l6  =  ±4V~l>  V—2$  =  ±*>  4/— I,  ..., 
and,  putting  z  for  |/—  1,  we  get  the  scheme 
•  •  •  —  Sh   —  Ah    —  3*>    -  2h    —  h  o,   i,   2/,   3*;  4*,   5*.  .  .  . 

We  call  the  new  numbers  imaginaries,  non-reals,  or  {-num- 
bers, while  other  numbers  are  real  or  /Z0;z-z'. 

Of  course  we  have,  in  the  same  way,  z-n umbers  where  the 
multipliers  of  i  are  fractional  or  incommensurable  ;  but,  for  the 
present,  we  confine  our  attention  to  those  with  integral  multi- 
pliers. We  call  them  integral  i-numbers,  and  say  that  their 
absolute  value  is  the  absolute  value  of  the  integral  multipliers, 
thus  making  of  i  a  new  unit. 

The  distributive  law  for  multiplication  gives  ai-\-bi  =  (a-\-b)i, 
which  defines  the  addition  of  z-numbers. 

64 


DOUBLE  NUMBERS  AND    THE   SIMPLER   OPERATIONS.      65 
For  multiplication,  we  have 

ai  X  bi  X  ci  =  |/—  a2  X  \f—  &  X  \/~  *  =  V—  a2--  b2>  —  c* 
—  V— **(-£*.  — **)=  |/—  b2--c2>  —a2=  V—a'-P-c2-  —  !-  —  ! 
=  —  1  •  V^a2!??. 

Consequently,  *w  X  &'  X  «  =  «*  X  (£z  X  ci)  =  bi  X  «  X  ai 
=  —  ^a,  and  the  commutative  and  associative  laws  hold  for 
the  multiplication  of  /-numbers. 

Further,  the  absolute  value  of  the  product  is  the  product  of 
the  absolute  values  of  the  factors  ;  the  product  is  an  i  or  non-i 
number  according  as  there  are  an  odd  or  an  even  number  of 
i- factors  ;  and  in  determining  its  sign,  each  pair  of  i-f actors  gives 
a  minus  sign  in  addition  to  the  minus  signs  before  the  several 
factors. 

Since  i  X  i  =  —  1,  i  X  —  i  =  1,  and  —  i  =  -.     Therefore 

an  z-n umber  occurring  as  a  divisor  in  any  expression  gives  an 
opposite  sign  to  that  given  by  an  /-number  occurring  as  a 
multiplier. 

The  raising  of  /-numbers  to  integral  non-z  powers  is  in- 
cluded in  the  rules  for  multiplication  and  division.  Nor  is 
there  any  difficulty  in  taking  integral  non-z  roots.  As  frac- 
tional powers  are  merely  integral  powers  of  integral  roots,  these 
are  similarly  disposed  of. 

Find  single  i  or  non-z  numbers  equivalent  to  the  following 
expressions : 

8/  .  9i  -4-  (i  .  2i  .  3z),  (2i)2  +  (3*>  +  8  +  (fi  -  Ai)\ 

ji-  (2ty+ ($if-i'\  i/i,  y%,  Vh  I/-64,  v-h 

».  (-*')*>  (-4)^,  (-25)1,  (-81)2,  ,-f,   |-|,  ,"-*, 

ai  .  bi  -4-  ci  .  di  -r-  ei  .  fi. 


66         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

100.  To  give  a  meaning  to  /-numbers,  think  of  our  defini- 
tion of  ordinary  positive  integers,  names  arbitrarily  given  to 
objects  when  counting  a  group  of  them. 

Suppose  we  have  several  groups  of  objects.  Number  the 
groups  as  well  as  the  objects  in  the  groups.  We  then  desig- 
nate any  object  by  two  numbers :  we  say  it  is  the  object  2  in 
the  group  3,  the  object  4  in  the  group  5,  and  so  on.  In  other 
words,  to  an  object  riumber  we  add  a  group  number ;  call  this  last 
an  /-number,  and  we  then  say  that  the  third  object  in  the 
fourth  group  is  the  object  3  -f-  4*- 

All  the  objects  3  in  all  the  groups  themselves  form  a  group; 
viz.,  the  group 

3  +  i,     3  +  2*,     3  +  3*,     3  +  41,  .  .  . 

0 

In  that  group  3  +  41  is  the  fourth  object,  and  we  are  justified 
in  writing  3  -f-  4/  —  4/  +  3. 

Of  course,  we  can  name  both  groups  and  objects  by  nega- 
tive numbers  as  well  as  positive,  and  we  can  have  starting  or 
nul  groups  and  objects. 

The  expression  a  -f-  bi  we  now  call  a  complex  or  double 
number,  of  which  a  and  b  are  the  non-i  and  i parts. 

To  get  to  3  +4/  we  count  forward  from  o,  the  name  of  the 
nul  object  in  the  nul  group,  3  object  numbers,  4  group  num- 
bers. To  get  (3  -|-  4*)  -f-  (5  +  70  we  count  from  either  double 
number  as  we  would  count  from  zero  to  get  the  other. 

.-.  (3  +  4O  +  (5  +  7i)  =  5  +  70  +  (3  +  40  =  8  +  1  ti- 

We  see  then  that  two  double  numbers  are  equal  if  their  i  and 
non-i  parts  are  separately  equal,  and  the  sum  of  any  number  of 
double  numbers  is  that  double  number  whose  i  and  non-i  parts  are 
respectively  the  sums  of  the  i  and  non-i  parts  of  the  double  num- 
bers added. 

Ordinary  or  non-z  numbers  are  merely  double  numbers 
whose  i  parts  are  zero ;  while  /-numbers  are  double  numbers 


DOUBLE  NUMBERS  AND    THE   SIMPLER   OPERATIONS.      67 

-whose  non-z  parts  are  zero.  In  symbols,  a  —  a  -f-  oi,  and 
ai  =  o  -f-  ai. 

Plainly,  all  the  laws  of  addition,  previously  established, 
continue  to  hold. 

101.  Assuming  multiplication  distributive  with  regard  to 
addition,  and  multiplying  1  +  2f  by  all  double  numbers,  we 
get  products  that  can  be  arranged  as  below. 


1-3*',       2-  i,       3+  i,       4+3*,       5+  Sh       6+  jit        7+  gt,  .  .  . 
•1  —  2*,       o        ,        1+2*',        2+4?,        3+  6*,       4+  8z,        5+io^»  •  •  • 


—  3—  i,  —2+  *,  -1+3A              5*.        !+  7*.       2+  9*,  3-\-ni,  .  .  . 

—  5        ,  —4+2/,  —3+4/,    —2+6*',    —1+  8z,             io*,  1+12*,  .  .  . 

-7+*,  -6+3/,  -5+5*.    -4+7?',    -3+9*j    -2+11/,  -1+13*,  •  • 

—9+2/,  —8+4*,  —7+6*',    —6+8*',    -5+10*',    —4+12*,  —3+14*,  .  .  . 


Here  the  product  (1  +-  2*)(3  +  4*)  is  2  -f-  II*  in  the  3d  row 
down  and  4th  column  right  from  o. 

We  have,  in  fact,  a  1  +-  21  arrangement  or  system  in  which 
the  product  is  the  3d  number  in  the  4th  group.  Thus,  to  mul- 
tiply a  number  a  +-  bi,  by  another  c  -\-  di,  we  pick  out  the  £th 
number  in  the  dth  group  of  the  a  -f-  bi  system,  or  what  is  the 
same  thing,  the  #th  number  in  the  b\\\  group  of  the  c  -\-  di 
system. 

It  is  as  though  we  spoke  of  6,  the  product  of  2  and  3,  as  the 
second  number  in  the  third  pair  of  numbers,  or  as  the  third 
number  in  the  second  triplet  of  numbers ; 

12,  34,  56,     or     123,  456. 

The  significance  of  all  this  will  become  more  apparent  on 
adopting  a  simple  geometric  representation. 


68 


AN  INTRODUCTION   TO    THE   LOGIC  OF  ALGEBRA, 


Suppose  the  objects  to  be  dots,  and  the  groups  rows  of 
them.     Thus : 


•  • 

•  • 

2i(3+i) 
©          •          .           .  \      • 

•          •••••• 

...•©.. 

oi                                                           ^ 

(-2+i) 
eX 

(3+i) 

•         ©          •           •          •  \ 

-2+<       *      \ 

u 

-7 

6-5-4-3        -2        -X — *" 

-2(3+ <)* 


.0123456 

-i\ ( 

-2i\  •          •          •         ®         •          • 

-Si    \-i(3  +  i) 

• 

. 



-At       •              Q)              •                *                •               • 

~W              -2i(3+<) 

• 

All  the  dots  belong  to  the  original  or  standard  I  +  oi  sys- 
tem ;  those  encircled,  to  the  3  +  i  system.  We  have  indicated 
the  dots  —3+i  and  (—  2  +  i)($  +  i). 

The  student  may  in  like  manner  construct  a  diagram  to 
show  the  i-\-i  system,  the  —  l  -\-  i  system,  the  2  -f-  oi  or  2 
system,  the  o  +  3^  or  3/  system.  Why  do  the  dots  of  the  first 
of  these  systems  coincide  with  those  of  the  second  ?  Why  do 
all  dots  belonging  to  the  3  —  i  and  2z  —  7  systems,  belong  also 
to  the  —  19  —  i  system?  Why  can  no  others  be  in  that 
system  ? 

The  commutative  law  for  the  multiplication  of  double 
numbers  is  contained  in  the  definition  of  a  product.  The 
student  can  show  that  the  associative  law  also  holds,  and  that 
multiplication  is  distributive  with  regard  to  addition  when  all 
the  numbers  involved  are  double. 


NON-INTEGRAL  DOUBLE  NUMBERS  :  TENSORS  AND  SORTS.    6$ 

Raising  to  integral  non-z  powers  is  done  by  repeated  mul- 
tiplication and  does  not  require  special  consideration.  Let  the 
student  construct  diagrams  showing  the  2  +  z,  (2  +  02>  anc* 
(2  +  z)3  systems ;  also,  the  z,  z2,  z3,  and  z4  systems.  In  what  do 
these  last  differ? 

102.  Of  course,  subtraction  is  a  mere  addition  of  opposites, 
while  division  is  a  guessing  what  to  multiply  one  double  num- 
ber by  to  get  another.  It  may  or  may  not  be  integrably 
possible. 

E.g.,     (i  +  3i)  +  (2+i)=i  +  t;    for    (1+0(2  +  0=  * +  3* 

But  there  is  no  double  number  x  +  iy,  with  integers  for  x 
and  y,  such  that  (1  +  i){x  -\-iy)  —  2-\-  i. 
For  expanding,  we  have 

x  —  y-\-(x-\-y)i  —  2-\-i    and    .\x—y  =  2    and    x-\-y=i; 

equations  which  no  integers  satisfy. 

Thus,  2  -{-  t  is  not  integrably  divisible  by  I  +  i. 

Similar  remarks  apply  to  the  extraction  of  integral  non-z 
roots. 

(1+  50-(3  +  20  =  ?  (1  +  70-(2  -  0=  ?  (4  +  9)-5-(2-30  -  ? 

25-*-(3  +  40=?  i7-(^-4)=?  13-^  +  3)=?  fS+4*==? 


V13  —  121  =  ? 


Show  that       1/(26  —  ioz)3  =  (  V26  —  ioz)3. 
Prove  the  following  not  integrally  possible : 
(1+41)^(2-20;   (3-0^(2  +  30;   (7+0-K5-0- 

II.    Non-integral  Double  Numbers  :  Tensors  and  Sorts. 

103.  Fractions  enter  as  with  ordinary  numbers.  There,  it 
will  be  remembered,  they  were  numbers  lying  somehow  between 
numbers  already  used.  So  double  numbers  with  fractional 
parts  lie  somehow  between  the  double  integers.     Better,  the 


JO         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

fractional  double  numbers  are  such  that  their  i  and  non-z  parts 
lie  between  the  i  and  non-z  parts  of  integral  double  numbers. 
They  are,  if  you  please,  names  assigned  to  new  objects  inter- 
polated into  our  groups  and  new  groups  interpolated  into  our 
systems. 

Thus,  in  our  diagram  above,  a  dot  placed  between  I  and  2 
on  the  initial  line  of  the  standard  system  would  be  marked  by 
one-and-a-fraction  ;  while  a  row  of  dots  lying  between  the  i  and 
the  2z  row  would  be  a  one-and-a-fraction  row. 

Manifestly,  the  interpolation  can  be  carried  to  any  extent, 
and  the  position  of  any  point  on  the  diagram  marked  as  accu- 
rately as  you  please. 

Let  it  then  be  carried  out.  Make  the  line  of  x's  of  §  96  the 
initial  line,  or  line  of  non-z  numbers ;  and  the  line  of  y's  there, 
the  line  of  /-numbers  here.  Finally,  represent  numbers  with 
equal  z  parts  by  points  equally  distant,  the  same  way,  from  the 
non-z  line ;  and  numbers  whose  non-z  parts  are  equal  by  num- 
bers equally  distant,  the  same  way,  from  the  z-line. 

In  brief,  put  (x,  y)  =  x  -\-  iy. 

We  have  a  representation  of  double  numbers  first  used  by 
a  French  mathematician,  Argand,  and  frequently  referred  to  as 
the  Argand  diagram. 

Evidently,  now  as  heretofore,  two  double  numbers  are  equal 
if  their  i  and  non-i  parts  are  separately  equal ;  and  one  double 
number  lies  between  two  other  double  numbers,  if  its  i  and  non-i 
parts  lie  between  the  i  and  non-i  parts  of  those  others. 

104.  Consider  the  number  7-  -f-  -j*«     This  is 

Thus,  all  fractional  double  numbers  are  simple  fractions  of 
double  integers. 

In  the   ad  +  bci  system,   j  -f-  -yi  belongs    to    the   initial 


NON-INTEGRAL  DOUBLE  NUMBERS:  TENSORS  AND  SOR  TS.    J I 

group,  and  the  general  type  of  all  numbers  belonging  to  that 
group  is  -7,  {ad  -f-  bci). 

Two  numbers  are  said  to  be  of  the  same  sort  if  they  belong 

e 
to  the  same  group  and  if  the  multiplier  -7.  has  the  same  sign 

for  both ;    but  of  opposite  sorts  if  the  number  -7  has  opposite 

signs  for  the  two.     They  are  of  different  sorts  if  not  belonging 
to  the  same  group. 

Show  that  t  +  -%i  and  —  -|-  ~-d  are  of  the  same  or  opposite 

sorts  according  as  b  does  or  does  not  agree  in  sign  with  c. 

_,  ,  t  r    (  tne  same  sort  )    , 

Show  that  numbers  of  i  ,A  r  have  their  points 

(.  opposite  sorts  ;  r 

on  the  Argand  diagram  co-linear  with  the  origin,  and  lying  on 

{"the  same  side  )      „     ' 
C  of  the  origin, 
opposite  sides  J  & 

105.  The  product  of  a  +  bi  by  c  -f-  di  is  ac  —  bd  -f-  {ad-\-  bc)i ; 
and,  obviously,  the  product  of  any  number  of  the  sort  a  -\-  bi 
by  one  of  the  sort  c  -\-  di  will  give  one  of  the  sort  ac  —  bd 
-\-{ad-\-  bc)i. 

Suppose  we  were  to  divide  a  -f-  bi  by  c  +  di  and  get  the 
result  x -\- yi.    Then 

a  +  bi  —  (x  +  yi){f  +  di), 
and 

(0  +  bi)(c  -  di)  =  {x  +  yi){c  +  di){c  -  di)  =  (x  +yi)(c2  +  d>) ; 
whence    ;r  +^' ==  (a  4~  ^Qfe  "~  ^0  "*■  (^  "H  ^2)* 
But  also  #  +^"  =  (^  +  ^0  +  fc  +  ^0» 

Now  c  -\-  di  and  <:  —  di  are  merely  two  double  numbers  that 
agree  in  everything  save  the  signs  of  the   /-parts.     We  call 


J 2         AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

such  numbers  conjugates,  and  we  have  just  proved  that  to  mul- 
tiply by  either  of  two  conjugates  gives  a  number  of  the  same  sort 
as  to  divide  by  the  other :  a  remarkable  and  very  useful  theorem. 

The  number  -|-  ^(non-z  part)2  -\-  (z'-part)2,  by  which  if  both 
of  two  conjugates  are  divided  the  results  are  reciprocals,  is 
called  the  modulus,  absolute  value,  or  tensor  of  the  numbers. 

The  student  will  see  that  when  either  the  z-part  is  zero  or 
the  non-z  part  is  zero,  this  agrees  with  previous  definitions  of 
absolute  value. 

Show  that  numbers  whose  tensors  are  equal  have  their 
points  on  the  Argand  diagram  equidistant  from  the  origin. 

106.  Let  a  -f-  ib  be  any  number,  and  write  -f-  ^d2  +  b2  =  m. 

_  a  b 

Further,  let  —  =  p  and  —  =  q.     Then 
m  m       * 

a-\-ib  =  m(p  -\-  iq),     and    p2  -\-q2  =  I . 

The  number  is  thus  resolved  into  two  factors :  a  quantity 
factor  determining  the  absolute  value,  and  a  quality  factor 
determining  the  sort.  The  latter,  whose  tensor  is  unity,  we 
call  a  complex  unit. 

Let  there  be  a  second  number  a  -f-  ib'  =  m' \p'  -f-  iq'),  and 
consider  the  tensor  of  the  sum,  of  the  difference,  of  the  product, 
and  of  the  quotient  of  the  two. 

107.  For  the  sum  we  have 


a  +  a'  +  Up  +  bf),   with  the  tensor    V{a  +  a')2  +  (b  +  by. 
The  tensor  squared  is 

a2  +  b2  +  a'2  +  b'2  +  2{ad  +  W)  =  m2  +  m'2  +  2mm'  (pp'  +  qq). 

Were  pp'  -j-  qq'  =  1 ,  this  would  plainly  be  the  square  of  m  -f-  m' ; 
and  we  should  have  the  sum  of  the  tensors  for  the  tensor  of 
the  sum.     This  does  happen  when/  =  p'  and  q  =  q' ;  that  is, 


NON-INTEGRAL  DOUBLE  NUMBERS:  TENSORS  AND  SORTS.    73 

when  a  +  ib  and  a  -\-  ib'  are  numbers  of  the  same  sort.  Other- 
wise pp'  -\-  qq'  <  I ,  and  the  tensor  of  the  sum  is  less  than  the  sum 
of  the  tensors. 

To  see  this,  take  account  of  the  relations  P2-\-q2=p'2  -\-q'2  =  I, 
and  write  pp'  +  qq  =  k.     Evidently, 

/2  _  2pp>  +p*  +  q*-  2qq'  +  a'2-2-2k; 

or  {p-pJ  +  {q-q'Y  =  2{i  -k). 

The  expression  on  the  left  is  the  sum  of  two  squares,  and 
positive,  unless/  =  p'  and  q  =  q' ; 

.'.  i  —  k>  o,     and     k  —pp'  -\-qq'  <  I. 

On  the  other  hand,         k  >  —  I. 

For       0<(p+py  +  (q  +  qJ  =  2(i+k),     and     i+£>o. 

/.  the  tensor  of  the  sum  is  greater  than  the  difference  of  the 
tensors  unless  p  —  —  p'  and  q  =  —  q' ' ;  that  is,  unless  a  -f-  ib 
and  a'  +  ib'  are  numbers  of  opposite  sorts. 

Show  that  the  tensor  of  the   {  -.-  r  of  two  numbers 

(  difference  ) 

vanishes  when  and  only  when  the  numbers  are   \  ,     \  , 

J  \     equal     ) 

Show  that  the  tensor  of  the  difference  of  two  numbers  ex- 
ceeds the  difference  of  their  tensors,  unless  the  numbers  are  of 
the  same  sort ;  while  it  is  less  than  the  sum  of  their  tensors, 
unless  the  numbers  are  of  opposite  sorts. 

Show  that  if  two  numbers  are  neither  of  the  same  nor  of 
opposite  sorts,  their  sum  can  be  neither  of  the  same  sort  as 
either  of  them  nor  of  an  opposite  sort  from  either. 


74         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

108.  The  tensor  of  a  product  is  a  simpler  matter.    We  have 

(a  +  ib)(a'  +  ib')  =  aa'  -  bb'  +  i(ab'  +  a'b) 

=  mm'  ipp'  -  qq'  +  i(pq'  +/?)]. 
But  {ppf  -  qq')2  =  fp'2  -  2Pp'qq'  +  q2q'\ 

and  (pq'  +p'q)2  =  fq'2  +  2pp'qq'  +  2p'2q2 ; 

and  so 

W-ssy-HPs'  +?'?Y  =Af  +  4'2)+<?(J>'2+2'2)  =f+<?= i. 

Thus  (pp'  —  qq;) -\-  i{pq'  -\-  p'4)  is  a  complex  unit,  and 
{a  -f-  ib){a!  -f-  ib')  =  mm'  X  that  complex  unit.  In  words  :  the 
tensor  of  the  product  of  two  numbers  is  the  product  of  their  tensors. 

It  goes  without  saying  that  the  ratio  of  the  tensors  of  two 
numbers  is  the  tensor  of  the  ratio  of  the  numbers. 

Let  the  student  prove  these. 

The  tensor  of  an  integral  power  of  a  number  is  the  integral 
power  of  the  number  s  tensor. 

If  there  be  an  integral  root,  its  tensor  is  the  integral  root  of 
the  number's  tensor, 

p      t 

p  lit 

(a  +  ibl(a?+~ib)/    v        ' 


M(a+ib) 


la'+ib  i 


2i 

ib 


4  ^-■•a+ib 


1  -— —' 


o  a'   i  aji  ^^ 

Use  the  diagram    representation.     We   go,  from  the  nul- 
point,  a  right  b  up  to  a-\-  ib  or  A.    Evidently  m,  =  +  Va2-\-  b2, 


NON-INTEGRAL  DOUBLE  NUMBERS:  TENSORS  AND  SORTS.    7$ 

is  the  distance  from  the  nul-point  to  A.  Likewise  m'  is  the 
distance  from  the  nul-point  to  A'  \  while  V(a  -\-  a')2  -j-  (b  -\-  b')2 
is  the  distance  from  the  nul-point  to  {a  -f-  id)  -\-(a'  -f-  ib')  or  S; 
o,  A,  A',  and  5  are  corners  of  a  parallelogram;  and  of  course 
oA+AS>  AS>  oA  -AS. 

Show  that  the  distance  AA'  is  V(a  —  a')2  +  (b  —  b')\ 
To  get  the  product   point  P,  we  go   from   the   nul-point 
{a'  -f-  #')-ward  the  distance  am\  and  then  i(a!  -{-  2^')-ward  the 
distance  bm' .     Thus  the  distance  of  P  from  the  nul-point  is 

Va2m,2  +  b2i?i'2  =  mm'. 

Prove  that  the  triangles  o,  I,  A  and  oA'p  are  similar. 

Construct  the  sum  and  product  points  if  a -{-id  and  a!  -{-ib 
are  interchanged. 

Construct  the  two  difference  points  and  the  two  ratio  points* 

Construct  (a  +  ib)2,  (a  +  iUf,  (a  +  ib)\  (a  +  ib)~\  {a  +  ib)~2 
(only  one  figure  is  necessary). 

The  problems  are  easily  varied  by  changing  the  numbers 
a,  a',  b,  and  b' . 

109.  We  have  said  that  one  double  number  lay  between 
two  others  when  the  i  and  non-2  parts  of  the  one  number  lay 
between  the  i  and  non-2  parts  of  the  two  others. 

Instead  of  referring  the  numbers  to  the  standard  system, 
refer  them  to  a  (/  +  iq)  system  {p2  -f  <f  =  1).  In  the  place  of 
non-2  and  i  parts,  we  now  have/  -|-  qi  and  ip  —  q  parts. 

Consistency  requires  that  the  definition  of  lying  between 
shall  be  extended  so  as  to  include  a  reference  to  any  possible 
system. 

We  say  then  that  one  number  lies  between  two  others  ifp  and 
q  can  be  so  chosen  that  the  p  -f-  iq  and  ip  —  q  parts  of  the  one 
number  shall  lie  between  the  p  -j-  iq  and  ip  —  q  parts  of  the  other 
numbers. 

Thus,  consider  the  numbers 

1+2,     4  +  32,     6  +  22. 
Although  3  does  not  lie  between  1  and  2,  yet  4  +  32  does 


j6         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

lie  between  i  -f-  *  ar*d  6  -J-  2z.     For,  refer  the  three  numbers  to 
a  f  +  *i  system.     They  become  respectively 

4(i+4)-K4-*)>  Vd+^-M-f),  V(l+4)-Wt-l) 

and         |  <  y  <  y,     while  also     -  J  >  -  J  >  —  y. 

To  get  clear  ideas,  let  the  student  plot  the  above  points  on 
the  Argand  diagram,  and  then  through  the  l  -\-  i  and  6  -|-  2# 
points  draw  lines  parallel  to  the  initial  and  i  lines  of  both  the 
standard  and  the  f  -|-  if  system.  He  will  thus  get  two  rectan- 
gles having  I  -f-  i  and  6  -j-  2z  for  opposite  corners.  The  rect- 
angle that  has  a  side  parallel  to  the  initial  line  of  the  standard 
system  does  not  contain  the  point  4  -f-  3*,  while  the  other  rect- 
angle does. 

To  try  every  possible  system  to  see  whether  one  number 
lay  between  two  others  would  be  tedious.  Take  three  num- 
bers, a%  -f-  ibt ,  a  -j-  ib,  a2  +  ib2.  If  a  does  lie  between  ax  and  a2, 
and  also  b  between  bl  and  b2 ,  no  test  is  necessary.  Putting  this 
in  a  slightly  different  form,  if  (ax  —  a){a  —  a2)  __  o,  and  also 
(bt  —  b)(b  —  b2)  2l  o,  then  a  —  ib  does  lie  between  the  other 
two  numbers.  Our  problem  then  is  to  find  a  condition  for 
lying  between  when  either  or  both  of  the  above  products  are 
negative. 

Whatever  the  complex  unit/  +  iq,  we  must  have 

ax  +  tbx  -  (Ja,  +  qb,)(P  +  iq)  +  (pbx  -  qax){ip  -  q\ 
a  +  ib  =  (pa  +  qb)(p  +  iq)  +  (jb  -  qa)(ip  -  q), 
a2  +  ib2  =  (pa2  +  qb2)(p  +  iq)  +  (pb2  -  qa2)(ip  -  q). 

The  multipliers  of  /+  *4  anc^  *P  ~  9  now  ta^e  the  place  °f 
al9  bs,  a,  b,  a2,  and  b2,  above.     It  therefore  follows  that  a  +  ib 


NON-INTEGRAL  DOUBLE  NUMBERS:  TENSORS  AND  SORTS.    77 

does  lie  between  a1  +  ibx  and  a2  ~\-  ib2  if,  and  only  if,  /  -f-  iq  can 
be  so  chosen  that 

[K  -  a)p  +  (bx  -  %][>  -  a2)p  +  (b  -  b2)q~\  =  o, 
say  I  *  II  =  O, 

and     [(£,  -  b)p  -  (ax  —  a)q][{b  —  b2)p  -  (a  -  a2)q\  =  o, 
say  III  X  IV  —  o. 

Now,  by  the  help  of  the  relation  p2  +  q2  =  I,  it  will  be  found 
that 

I2+II2+III2+lV2=:(^I-^)2+(^-^)2+(^-^2)2+(^-^)2  =  A  say  ; 
and         (I  +  II)2  +  (HI  +  IV)2  =  (a,  -  a2J  +  (/;,-  b2\  =  B  say. 

If  B  <  A,  I  X  n  +  in  X  IV  <  o,  and  certainly  either 
I  X  n  <  o,  or  else  III  X  IV  <  o. 

But  B  <  A  unless  (a,  —  d)(a  —  a2)  +  (bx  —  b)(b  —  b2)  =  o. 

.*.  a  -\-  ib  does  not  lie  between  ax  -f-  ibx  and  a2  -f-  ib2  except 
under  the  same  condition. 

Thus,  2  +4/  does  not  lie  between  1+/  and  4  +  22;  for 
(-0(- 2) +  (-  3)2  = -4<o. 

To  show  that,  if  (at  —  d){a  —  a2)  +  (bs  —  b)(b  —  b2  —  o,  p-{-  iq 
can  be  so  chosen  that  neither  I  X  II  nor  III  X  IV  shall  be  nega- 
tive is  easy.  We  merely  chose  it  so  that  I,  say,  =  o.  This 
makes  I  X  II  =  o,  and 

because  I  X  n  +  in  X  IV  =  o,  III  X  IV  =  o. 

^,  ,      1         r  •    •      bx  —  b  —  i{al  —  a) 

The  required  value  of  p  -\-  iq  is 


v^-by  +  i^-ay 

Notice  that  B  above  is  the  tensor  squared  of  (ax  -f-  ibx) 
—  (a2  -f-  ib2),  while  A  is  the  sum  of  the  squares  of  the  tensors 
of  L  -j-  ibx)  —  {a  +  iff)  and  {a  +  ib)  -  (a2  +  ib2).  That  B  shall 
then  be  less  than  A  means,  on  the  Argand  diagram,  that  a  +  ib 
shall  be  somewhere  within  a  circle  constructed  on  the  junction 
of  ax  -f-  ibs  and  a2  -\-  ib2  as  a  diameter. 


7$         AN  INTRODUCTION   TO    THE   LOGIC  OF  ALGEBRA. 

1 10.  Suppose  at  -f-  tdl ,  a  -J-  ib,  and  «2  -f-  ib2  to  be  such  that 
I  X  n  =  o,  no  matter  what  the  value  of  /  -\-  iq.  Then 
(al  —  d)(a  —  a2)  ~  o,  and  also  (b1  —  b)(b  —  b2)  —  o,  as  can  be  seen 
by  putting  first  p  =  I  and  then  q  =  I  in  the  values  of  I  and  II. 
Furthermore,  since  p-\-iq  can  become  any  complex  unit,  it 
may  become  q  —  ip.     Now,  when  p  -f-  iq  changes  to  q  —  ip,  I  is 

changed  to  III  and  II  to  IV.    Consequently,  if  always  I  X  H  —  o, 

likewise  always  III  X  IV  =  o. 

Since  I  and  II  are  positive  together  and  negative  together, 
they  must  be  zero  together.  For,  let  p0  -f-  iq0  be  a  value  of 
J)  -f-  iq  that  makes  I  vanish. 

Then 

I  =  (al  —  d)p0  +  (b,  —  b)q0  =  o,  and  11  =  {a  —  a2)pQ  +  {b  —  b2)qQ. 

If  II  >  o,  let  p  -\-  iq  take  a  value  such  that  II  gets  smaller, 
b>ut  still  remains  larger  than  zero  (the  student  may  show  this 
possible). 

Because  the  signs  of  the  terms  of  I  agree  with  the  signs  of 
the  terms  of  II,  I  also  gets  smaller  and  therefore  negative. 
I  and  II  thus  cease  to  agree  in  sign,  and  we  could  not  have 
I  X  n  >  o.  Consequently  II  >  o  when  I  —  o  ;  neither  is  II  <  o. 
ax  —  a       —q0       a  —  a2 


We  have  then 


A 


a  condition  to  be  satisfied  by  a,  b,  alt  blt  a2  and  b2  in  order  that 
our  supposition  that  I  X  II,  and  so  III  X  IV,  should  never  be 
negative,  may  be  realized. 

We    say,  in    this   case,  that  a  -\-  ib   lies   directly   between 
ax  +  ibi  and  a2  -f-  ib2 . 

What  condition   besides  ■=- v  =  i r  must  be  fulfilled 

bl  —  b       b  —  b2 

in   order  that  a  -\-  ib  shall  lie  directly  between  a,  -f-  ibx  and 

a2  -f-  ib2?     What  that  a,  +  ibx  shall  lie  directly  between  a  -J-  ib 

and  a2  +  ib2?  that  a2  -\-  ib2  shall  lie  directly  between #  -f-  ib  and 

axAribJ 


COMPLEX   UNITS  AND  NON-i  POWERS.  79 

Show  that  one  complex  unit  cannot  lie  directly  between  two 
other  complex  units. 

Prove  the  following: 

If  one  double  number  lies  directly  between  two  other  double 
numbers,  then  a  number  directly  between  it  and  either  of  them 
lies  directly  between  those  two. 

If  a  number  lies  between  two  others,  but  not  directly  so,  a 
number  between  it  and  either  of  them  need  not  lie  between 
those  two. 

If  a  number  lies  between  two  others  \  .  £   directly 

(.  but  not  )  J 

so,  then  a  number  between  it  and  either  of  them,   ]  [ 

(     and     ) 

directly  so,  lies  between  the  two  numbers,  but  not  directly  so. 
Work  these  out  by  the  numerical  conditions  and  then  illus- 
trate on  the  Argand  diagram. 

III.    Complex  Units  and  Non-z  Powers. 

III.  In  §  108  we  proved  that  the  tensor  of  the  product  of 
two  numbers  was  the  product  of  the  tensors  of  the  numbers. 
It  followed  that  if  there  were  an  integral  non-z  root  of  a  double 
number,  its  tensor  was  the  integral  root  of  the  number's  tensor. 
We  now  consider  the  possibility  of  this  integral  root.  Evidently 
the  possibility  hangs  upon  whether  a  complex  unit  has  such  a 
root  or  not. 

For  simplicity,  think  of  the  square  root  and  suppose 


Then 


Vp  +  iq  =  x  +  iy. 
{x  +  if?  ==  P  +  *£»     and  so     x2  —  y2  =  p. 


But  T(p+iq)  =  i.  (T means  "tensor  of.") 

« 

.\  T(x  +  iy)  ==  I     and    x2  +  y2  =  i. 


80         AN  INTRODUCTION   TO    THE  LOGIC   OF  ALGEBRA. 

Thus  x=  ±  V—^--,  y  =  ±  V  l—J-->  and  for/  >  o,  ^  >  o, 

the  required  root  is  y 1-  i  y ,  or  the  opposite  thereof, 

as  may  easily  be  verified. 

By  repetitions  of  this  process  we  can,  of  course,  get  the 
fourth  root,  the  eighth,  any  root  whose  index  is  an  integral 
power  of  2. 

We  have  just  seen  that  if  both  p  and  q  are  positive  we  can 
get  a  square  root  both  of  whose  parts  are  positive.  But  no 
matter  what  the  complex  unit  with  which  we  start,  a  square 
root  can  be  gotten  whose  i  part  is  positive,  and  a  fourth  root 
both  of  whose  parts  are  positive.     Thus : 

Vp+iq     =+y—^--\-i\/l--^-i  and  so  on  ; 

Vp—iq     =—V  ~\-i\  ~ >  which  is  of  the  form  —p-\-iq ; 

V-p+lq=+V  1-J--  +  iV  l—~>  and  so  on; 

/i  —  p  /i  -\-p 

y—p—tq=  —  y \-iy  ,  which  is  of  the  form  —p-\-iq. 

i  -4-p 
112.  When/  and  q  are  both  positive,/2  < p  < ,  and 

therefore         p<y  »     while    q>  y . 

As  we  keep  on  extracting  doubly  positive  square  roots,  the 
i  part  finally  gets  as  near  as  you  please  to  zero  ;  while,  at  the 
same  time,  the  non-z'  part  gets  as  near  as  you  please  to  unity. 

Call  such  a  resulting  root  k  +  hi,  and  consider  the  series  of 
powers  k  +  hi,  (k  +  hi)\  (k  +  hi)\  . ',  .  to  (k  +  hi)n,  =p  +  iq. 

They  are  all  doubly  positive  complex  units.  Each  has  its 
i  part  greater  and  its  non-z'  part  less  than  that  of  the  one  before 
it.  The  tensor  of  the  difference  of  two  successive  powers  is 
constant  and  =  4/2(1  —  k). 


COMPLEX    UNITS  AND  NON-i  POWERS.  8 1 

To  see  the  truth  of  this  last  statement,  let  l-\-  mi  be  one  of 
the  series  of  powers.  The  next  one  is  {k  -f-  hi){l  '  +  mi),  and 
their  difference  is  (/  -\-  mi){k  -f-  hi  —  i).  Thus,  at  once,  the 
tensor  in  question  is 

7\l+  mi)  X  T{k  +  hi  -  1)  =  i  X  T(k  +  hi  -  i) 


=  V(i  -  k)2  +  h2  =  V2(i  -  k). 

It  is  plain  that  all  doubly  positive  numbers  r  -f  si  with  /part 
less  than  q  must  lie  between  two  powers  of  k  -|-  hi ;  for  the  i 
and  non-z  parts  of  the  number  will  lie  between  the  i  and  non-z* 
parts  of  two  of  the  powers.  (From  our  point  of  view,  coinciding 
with  either  is  merely  an  extreme  case  of  lying  between.)     Since 

(k  +  hi)  =  vp  -\-  qiy  we  can,  by  making  n  large,  make  //  as  near 
to  zero,  and  so  ^2{\  —  k)  as  near  to  zero,  as  you  please.  It  fol- 
lows that  r  +  si  is  hemmed  in  as  closely  as  you  please  by  a 
power  of  a  root  of  p  -\-  qi ;  i.e.,  r  -\-  si  can  be  expressed  as  closely 
as  you  please  by  a  fractional  power  of  p  -f-  qi;  and,  conversely, 
any  power  of  /  -\-qi  with  index  less  than  unity  but  greater  than 
zero  can  be  expressed  as  closely  as  you  please  by  a  number 
?  -\-  si. 

113.  We  shall  now  show  that  all  complex  units  whatsoever 
can  be  expressed  as  powers  oi p -\- qi,  and  that  to  every  power 
of  p -\- qi  corresponds  a  complex  number. 

In  order  to  do  this  we  establish  three  theorems : 

1st.  If  k  >  />  h  and  k  >  m  >  h,  then  the  complex  units 
l-\-  mi,  —  / -{-mi,  —  I  —  mi,  and  /—  mi,  will  on  multiplication 
by  k  -f-  hi  have  both  their  i  and  non-z'  parts  changed  by  more 
than  1  —  k,  but  by  less  than  //. 

Consider  the  first  of  the  four.     The  changes  in  question  are 

/  —  kl-\-  mh     and     km  -\-  hi  —  m. 

Now  /  -  kl  +  mh  >  1  —  k  if  /+  k  +  mh  >  kl  +  k2  -f-  h2, 
which  is  obviously  true  since  each  term  on  the  left  is  greater 
than  the  corresponding  term  on  the  right.     Similarly, 


82         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

km  +  hi  —  m  >  i  —  k,     because     km  -\-  hi  -f-  k  >  k2  -\-  h2  -f-  tn  ; 

I  —  kl-\-  nth  <  h,  because       kl  +  mh  <  I  -f-  h; 

and 
km  -\-  hi  —  m  <  ^,  because     km  -f-  ///  <  //z  -{-  ^. 

Since  no  distinction  is  made  in  the  conditions  imposed  upon 
/  and  m,  these  letters  can  be  interchanged  in  all  of  the  above  in- 
equalities without  destroying  their  truth.  But  it  will  be  found 
that  the  changes  produced  in  —  l-\-  mi,  —  / —  mi,  and  /+  mi 
on  multiplication  by  k  -\-  hi  will  either  be  the  same  as  the  above 
changes,  or  else  derivable  from  them  by  the  interchange  of  m 
and  /.  In  fact,  I  -\-  mi  itself,  save  for  the  interchange  of  m  and 
/,  is  converted  into  the  three  other  forms  by  merely  passing  to 
the  i,  i2,  and  i3  systems. 

2d.  If   I  ~  g  >  k  and  o  =  f  <  //,  then  the  complex  units 

f+gi,  —g+fi,  —  f '  —  giy  g  —  fi  become,  by  the  multiplica- 
tion by  k  +  hi,  units  —  /'  +g'i,  —g  —fi,  f  —  g'h  g'  +/% 
where  the  numbers^  and  g'  fulfil  the  conditions  I  >  g'  —  k, 

o<f'  =  k. 

As  before,  to  prove  this  for  one  of  the  given  complex  units 
is  to  prove  it  for  all.     We  have 

(f+gilk  +  hi)  =fk-  hg+  i{fh  +  kg). 

Now  fk  —  hg  is  largest  when  fk  is  largest,  and  smallest  when 
kg  is  largest :  is  largest,  therefore,  when  /is  nearest  h  and  g  is 
nearest  k,  and  smallest  when  f  •=  o  and  g  =  I.     Hence 

o  >  fk  —  hg  >  —  h,      or      o  <  hg  —  fk  <  h. 

As  for  fh  -\-  kg,  it  is  positive  and  largest  when  fk  —  hg  is  least 
in  absolute  value,  and  least  when/^  —  hg  is  largest  in  absolute 
value. 

...  Y  >//l  +  kg=k. 

Thus  hg  —  fk  and  fh  -\-  kg  can  be  replaced  by  numbers  f  and 
g'  conditioned  as  above. 


COMPLEX    UNITS  AND   NON-i   POWERS.  83 

3d.  If  the  complex  units  —  f  -\- g'i,  —  g'  —  f'i,  f  —  gi, 
g'  -{-f'i,  be  multiplied  by  k  +  hi,  we  shall  get  respectively 
numbers  —  /+  mi,  —  I  —  mi,  I  —  mi,  /+  *».  We  leave  the 
proof  for  the  student. 

Start  now  with  unity  or  (k-\-hi)°,  and  form  in  succession 
the  powers  k  +  hi,  (k  +  hi)2,  (k  +  hi)3,  .  .  .  We  shall  get,  in 
order,  numbers  I  -\-  mi,  a  number  /-{-gi,  a  number  —  f  -{-g'i, 
numbers  —  /  +  mi,  a  number  —  g -\- fi,  a  number  —  g'  —  f'i, 
numbers  —  I  —  mi,  a  number  —  /  —  gi,  a  number  f  —  g'i, 
numbers  /  —  mi,  a  number  g  —  fi,  a  number  g'  -{-f'i,  and 
finally  numbers  l-\-mi  again. 

Now  observe:  if  three  different  complex  units  agree  in  the 
sign  of  one  of  their  parts,  one  of  the  three  numbers  must  lie 
between  the  other  two. 

If  they  agree  in  the  signs  of  both  their  parts,  this  is  obvious 
enough.  E.g.,  any  number  I -\- mi  lies  between  some  two  of 
those  numbers  /-f-  mi  that  we  get  by  the  powering  of  k-{-  hi. 

If  the  three  agree  in  the  sign  of  one  of  their  parts  only,  the 
lying  between  may  not  be  obvious.  E.g.,  when  f>  f,  does 
a  number  f  -\- gi  lie  between  a  number  —  f  -{-g'i  and  a  num- 
ber I -\- mi}  Apply  the  test.  We  have  for  the  quantity  that 
is  not  to  be  less  than  zero 

If  this  expression  can  be  diminished  without  becoming  nega- 
tive, it  must  be  positive.  But  this  is  precisely  what  happens 
when  for  /we  write  f  and  for  —  m  we  write  -\-g'.  Each  term 
of  the  expression  is  diminished,  and  their  sum  becomes 
f ' 2  —  f2  +  g' 2  —  g2->  which  is  zero.  Similarly  can  be  treated 
other  sets  of  three  complex  units  agreeing  in  the  sign  of  only 
one  of  their  parts. 

Consequently  all  doubly  positive  complex  units  not  lying 
between  two  doubly  positive  powers  of  k  +  hi  must  lie  between 
the  last  doubly  positive  power  and  the  first  negative-positive 
power,  i.e.  between  the /"-J- ,£7  and  the  — f  -f-  g'i  number  due 
to  the  powering.  Similar  statements  apply  to  negative-posi- 
tive, doubly-negative,  and  positive  negative  units,  and  to  the 


84         AN  INTRODUCTION    TO    THE  LOGIC   OF  ALGEBRA. 

numbers  i,  —  I,  —  i,  and  I  of  which  the  non-expressed  parts  are 
ambiguous  in  sign.  Thus  the  powers  of  (k  -\-  hi)  hem  in  all 
complex  units  and  our  thesis  is  proved.  Of  course,  when  we 
say  "  hemmed  in,"  we  take  for  granted  that  h  shall  be  as  small 
as  you  please,  and  so  ^2(1  —  k),  the  tensor  of  the  difference  of 
the  two  successive  powers  between  which  any  assigned  com- 
plex unit  must  lie,  as  small  as  you  please. 

114.  We  have  shown  that  all  complex  units  whatsoever  are 
hemmed  in  by  powers  oi  k -\-  hi.  Because  (k  —  hi)  is  the  re- 
ciprocal of  k  -f-  hi,  and  every  complex  unit  is  the  reciprocal  of 
some  other  complex  unit,  all  complex  units  are  likewise 
hemmed  in  by  powers  of  k  —  hi.  For  the  successive  powers 
of*  k  —  hi,  the  i  and  non-z  parts  go  through  their  changes  in 
precisely  the  reverse  order  to  that  for  powers  of  k  -\-  hi.  In 
fact,  we  get,  in  order,  positive-negative  numbers,  doubly-nega- 
tive numbers,  negative-positive  numbers,  and  doubly-positive 
numbers. 

Why  can  we  not  hem  in  complex  units  by  successive  powers 
of  a  negative-positive  or  a  doubly-negative  number? 

115.  Because  all  complex  units  are,  as  near  as  one  pleases, 
powers  of  k  -|-  hi,  any  complex  unit  whatever  is  some  power  or 
other  of  any  other  complex  unit.     E.g., 

If  p  +  iq  =  {k-\-  hif  and  r-\-si  —  (k-\-  hi)11,  p  -\-  iq  —  (r  -J-  w)ft. 

Conversely,  a  complex  unit  can  be  found  which  comes  as 
near  as  you  please  to  any  assigned  power  of  any  given  complex 
unit  p  +  iq. 

m 

Suppose  we  want  (p  +  qi)~l>  If  ft  is  a  power  of  2,  no  expla- 
nation is  necessary.     If  not  a  power  of  2,  there  are  always  two 

fractions,  -y,  — — ,  differing  as  little  as  you  please,  such  that 

a       m      a-\- 1 

~¥<Ti<  _?~, 

m  •  a  a-f-i 

and  {p  -f-  iqfi  is  hemmed  in  by  (p  -\-  qi)rf  and  (p  -f-  gi)  a*  . 

Calculate,  for  instance,  (f  +  z'f-)"7\ 


COMPLEX    UNITS  AND  NON-i   POWERS. 


85 


We  have 

and 

Whence 

and 


V-  <^  ¥  < 


^1  +  *'i  =  °-9944  +  o.  10572. 

(!  +  '*)*  =  0.179  +  0.984/, 
(f  +  ^)v  =  0-074  +  0.997/. 


More  closely,  f«|-f  <  Jf  <  \Ui  i  and  we  find  that  (f  +  *f)V 
lies  between  0.1 126  +  0.9936/  and  0.1 135  +  0.9935/. 

Very  accurately,  (f  +  tftV  =  0.1 133739  +  0-9935537/. 
.  As  an  aid  to  clearness,  we  have  plotted  on  the  Argand  dia- 
gram the  points  (J-  +  iff,  (|  +  *$)*,  (|  +  iffl,  ...  to  (f  +  4)¥, 
marking  them  o,  1,  2,  .  .  .  to  55. 


.18. 


13.99%%  a©  . 


.10  - 


20. 


r 


23. 
24. 
2?i 
§6. 
•&. 

fc 
30- 
31. 


i     •■? 

!    • 


•5 


•  i_ 

.55 

To. 

•51 


83* 


3T  -. 


>so 


*&• 


_'47 


40   II    42     43 


It  will  be  noticed  that  1  is  at  o ;  /  is  between   13  and   14; 

—  1,  between  27  and  28  ;   —  /  between  40  and  41  ;  and  finally, 
1  is  also  between  54  and  55. 

Just  to  the  right  of  T3  is  the  dot  for  (f  +  iff*. 
If  we  call  I  +  /£ ,  k  -f-  hi,  then  o  to  12  are  the  /-f-  mi  powers, 
13  is  the/ +  £*  power,  14  the  —  /'  +  ig'  power,  FJ  to  26  the 

—  /  -f-  mi  powers,  and  so  on. 


86         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

The  dots  to  represent  the  successive  powers  of  (J-  -f-  t^)™1™ 
would  be  128  times  as  near  together  as  those  for  the  powers  of 
(I  ~~t"~  *'£)*>  ano-  could  not  be  readily  represented  on  our  scale. 

We  have  shown  how  any  non-z  power  of  a  complex  unit  can 
be  gotten  with  any  desired  degree  of  accuracy. 

Any  double  number  is  a  product  tensor  X  complex  unit;  and 
therefore,  any  power  of  a  double  number  is  the  product  power-of- 
tensor  X  power-of -complex-unit . 

Thus,  a  definite  meaning  is  given  to  all  non-z  powers  of 
double  numbers.  There  remain  for  consideration  the  i  and 
double-number  powers.  For  this  we  prepare  by  extending  to 
double  numbers  our  ideas  of  growth. 

IV.    Growths,  Rates,  and  Amounts. 

Il6.  A  double  number  grows  by  the  separate  growths  of 
its  i  and  non-z  parts.  These  growths  may  be  connected  by  any 
law.  In  particular,  they  may  be  so  connected  that  the  i  part 
grows  uniformly  with  regard  to  the  non-z  part. 

Thus,  if  x  +  iy,  where  y  =  ax,  grows  so  that  always  y  keeps 
equal  to  ax,  then  the  rate  of  growth  of  y  compared  to  x  is  a. 
If  x'  -\-  iy'  is  any  number  whatever  reached  by  the  growth,  then 
the  number  x  —  x'  -f-  i (y  —  y')  is  constantly  of  the  sort  1  -f-  icu 
We  say  that  the  growth  is  a  uniform  one  of  the  sort  1  -\-  ia. 

If,  on  the  other  hand,  the  rate  of  i  growth  compared  to 
noni-z  growth  is  not  uniform,  the  number  x  —  x'  -\-  z( y  —  y' ) 
constantly  changes  its  sort.  We  say  that  the  x  -f-  iy  growth  is 
of  a  varying  sort,  or,  more  briefly,  is  a  varying  growth. 

When,  for  instance,  x  -\-  iy  grows  so  that  always  y  =  x2,  the 
rate  of  i  growth  compared  to  non-z  growth  is  2x,  and  itself 
grows  with  x  at  the  rate  2.     The  sort  of  growth  at  x'  -f-  iy'  is 

given   by  x  —  x'  -\-  i(y  —  yf),   and    is    therefore    i-\-i- — ~ 

x  —  x' 

which,  for  y  =  y\  becomes   1  -f-  2x'i.     Dropping  accents,  the 

growth  at  x  -\-iy  is  of  the  sort  1  -f-  zxi,  a  number  that  changes 

with  changing  x. 

On  comparing  the  above  with  §  86  sequiter,  it  will  be  seen 


GROWTHS,    KATES,   AND  AMOUNTS.  87 

that  what  we  there  called  the  graphs  of  y  =  ax,  y  =  x2,  etc., 
are  merely  representations  of  the  growths  of  x  -\-  iy  according 
to  the  laws  y  =  ax,  y  =  x2,  etc.  Uniform  and  varying  growths 
are  represented  respectively  by  straight  lines  and  curves,  while 
the  sort  of  growth  at  a  point  x  -\-  iy  determines  the  law  of 
growth  of  a  tangent  to  the  graph  at  the  point  in  question. 

117.  Whatever  be  the  law  by  which  x  -\-  iy  grows,  we  can 
connect  with  it  the  growth  of  another  number  u  -\-  iv. 

Suppose  y  =  ax,  and  u  +  iv  =  (c  -\-  id)(x  +  iy). 

Then  u  -f-  iv  as  well  as  x  -\- iy  has  a  growth  of  a  uniform 
sort.  In  fact,  the  multiplication  by  c  -f-  id  merely  changes  the 
system,  so  that  the  growth  of  u  -f-  iv  would  be  represented  by 
a  straight  line  through  zero  and  (1  -f-  ia){c  -\-  id),  just  as  the 
growth  of  x-\-  iy  is  represented  by  a  line  through  zero  and  1  -f-  ia* 

The  growth  of  u  +  iv  is  thus  of  the  sort  c  —  ad-\-  i(d-\-  ac\ 

and  the  rate  of  i  growth  compared  to  non-2  in  u  -f-  iv  is  — —  _ 

c  —  ad 

The  rate  of  growth  of  u  -\-  iv  compared  to  x  -\-  iy  is  of  course 

u  4-  iv  —  (ur  4-  iv')  .    .  . 

— \ — i ^—7—1 — nf  =  c  4-  id,  a  constant. 

X+iy-  (x'  +  iy') 

This  last  result  is  independent  of  how  x  -\-iy  grows.  Does 
x-\-iy  grow  so  that  always  y  =  x2  ?  As  before,  the  rate  of 
u  -f-  iv  growth  compared  to  x  -f-  iy  growth  is  c  -f-  id,  and  the 
graph  of  n  -|-  iv  in  the  c-\-  id  system  is  the  same,  for  this  sys- 
tem, that  the  graph  of  x  -\-  iy  is  for  the  standard  system. 
These  graphs  are  sketched  in  the  annexed  diagram.  That  of 
x  -\-  iy  with  the  standard  reference-lines  is  drawn  full ;  that  of 
u  -\-  iv  with  the  transformed  reference-lines  broken.  The  arrows 
indicate  the  directions  of  positive  growth  in  the  two  systems. 

The  rate  of  ic  —  d  growth  compared  to  c  -\-  id  growth  in  the 
transformed  graph  is,  of  course,  that  of  i  compared  to  non-z' 
growth  in  the  original  graph,  viz.  2x. 

The  rate  of  i  with  regard  to  non-z'  growth  in  the  transformed 
graph  is 

u  —  u       ex2  -f-  dx  —  ex2  —  dx       2cx  +  d 


ex  —  dx2  —  ex  -j-  dx2       c  —  2dx 


88 


AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA, 


2(c+id) 


We  leave  it  to  the  student  to  find  the  rate  of  growth  of  v 
compared  to  x,  of  u  compared  to  y,  of  v  compared  to  y,  and  of 
u  compared  to  x. 

118.  From  the  two  numbers  a  +  ib  and  a'  -\-  ib',  respectively, 
let  there  be  uniform  growths  of  the  sorts  p-\-iq  and  /'  -f-  iq' • 
As  usual,  p2  +  q2  —  p'2  +  q'2  =  I- 

Any  number  reached  by  the  first  growth  can  be  written 
a  -f-  mp  -\-  i{b  -\-  mq)  ;  while  any  number  reached  by  the  second 
is  a!  +  m'p'  +  Up'  +  m'q'). 

If  now/+  iq  and  p'  -f-  iq'  are  not  of  the  same  or  opposite 
sorts,  we  can  always  find  one  and  only  one  pair  of  numbers 
m,  m'y  satisfying  the  condition 

a  -f-  mp  -fr  Up  -j-  m4)  —  a'  +  ;////>/  +  *  (^'  +  ^  V)- 
The  numbers,  in  fact,  are 


/(3_^_g>_fl0  g(a_a0_^_^ 

m  = : — --7 77 and     m  =   — -; - 

pq'-P'q  pq-pq 


GROWTHS,    RATES,    AND  AMOUNTS.  89 

In  case  m  =  o,  p'  +  iq'  is  of  the  sort  a  —  a  -\-i(b  —  b'),  and 

a  —  a!       b  —  b'  . 

= =  m  ;    statements  that  remain  true  when  ac- 

/  Q 

cented  and  unaccented  letters  are  interchanged. 

When  p  -\-  iq  and  /'  -f-  iq'  are  of  the  same  or  opposite  sorts, 
either  m  and  in'  are  non-finite  {in  =  00,  m'  =  00)  and  no  num- 
ber reached  by  the  one  growth  can  be  reached  by  the  other ;  or 
else  you  can  take  either  what  you  please,  provided  the  other  is 
rightly  paired  with  it. 

The  free  choice  is  possible  when  either  p -\-  iq  and  pr  -\-  iq' 

are  both   of  the  same  sort  as  a  —  a'  -f-  i{b  —  b')>  both  of  an 

opposite  sort  from  it,  or  one  of  them  of  the  same  sort  and  the 

o 
other  of  the  opposite  sort.    For  this  gives  m  =  — ,  or  o  X  in  =  o ; 

which  is  true  no  matter  what  number  m  may  be. 

Cl  —  Cl'  — (—  Dtp 

Having  taken  m.  we  have  for  m'  the  value , . 

■r     ,.1  /      1  .....  a '  —  a  +  m'p' 

In  like  manner,  m   taken  arbitrarily  gives  m  = . 

P 

The  distinction  between  growths  of  the  same  and  opposite 
sorts  can  of  course  be  avoided  by  agreeing  that  a  growth  of  an 
opposite  sort  is  merely  one  of  the  same  sort  taken  negatively. 

119.  The  preceding  investigation  furnishes  a  simple  test  for 
one  number's  lying  directly  between  two  others. 

That  p  -f-  iq  and  p'  -f-  iq'  shall  each  be  of  the  sort 
a  —  a'  -\-  i(b  —  b')  is  the  same  as  saying  that,  if  x  -\-  iy  be  the 
number  reached  by  the  growths,  then 

b  —  y     y  —  b' 


a  —  x      x  —  a 


Now  x  -j-  iy  is  directly  between  a  -\-  ib  and  a'  -\-  ib'  if,  be- 
sides this,  b  —  y  and  y  —  b'  agree  in  sign. 

But  x  +  iy  =  ci  +  ib  +  m{p  -\-  iq) 

,    ...  g-a'+fy-y) 


90         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 


which,  if  we  put  k  for  m  -—  Via—a')2  -j-  (b  —  b'f,  becomes 

(i  +  k)(a  +  ib)  -  k[a'  -f  ib'). 

Thus  y  =  {i  +  k)b-kb'; 

whence  b  —  y  =  k{b  -  b'\     and    y  —  b'  =  (i  +  k){  b-  V). 

That  b  —  y  and  y  —  br  should  agree  in  sign  it  is  therefore 
necessary  and  sufficient  that 

k{i  +  k)  <  o ;  i.e.,  —  i  <  k  <  o. 

All  the  numbers,  then,  directly  between  a  4-  ib  and  #'  +  itf 
are  contained  in  the  form  l(a  -\-  ib)  -\-  I' (a1  -f-  /^'),  where  /  and  /' 
are  positive  numbers  whose  sum  is  unity. 

Furthermore,  as  the  student  can  easily  show,  if,  keeping 
/-)-  /'  =  i,  we  take  /'  negative,  a  -f-  ib  lies  directly  between 
a'  -f-  ib'  and  the  number  that  we  get ;  while,  if  we  take  /  nega- 
tive, a'  -f-  ib'  is  the  number  between  the  other  two. 

From  7  +  8/  and  2  —  31,  respectively,  are  growths  of  the 
sorts  f  —  i\  and  —  yf  + 1 '-fp  What  is  the  number  reached  by 
both? 

From  the  same  numbers,  by  growths  of  what  sorts  would 
8  —  lyi be  reached ? 

Determine  g  in  the  following  numbers  so  that  they  shall  lie 
directly  between  7  -f-  Si  and  —  13  +  42z: 

g+211,     —7+&>     *>—gh     Zoi-g. 

Represent  these  problems  on  the  Argand  diagram. 

120.  The  number  a  +  ib  is  any  number  whatsoever.  So 
also  is  a  -f-  mp  +  M  ~\~  m(2)-  ^n  order  that  it  shall  be,  say, 
c  -f-  idy  we  require  simply  «  -f-  mp  =  c  and  b  -\- mq  =  d;  whence 


d-b 
m  = 


c-a  /(d-b)2-\-(c-a)2        / 

—p~  =  y  J+f "  V(d-b)2+(c-ay 


GROWTHS,   RATES,    AND  AMOUNTS.  9 1 

Thus  «r,  /,  and  q  are  given  in  terms  of  a,  b,  c>  and  d,  with 
the  possibility  of  always  taking  m  positive. 
In  other  words,  we  can  always  have 

m  =  T  [c  +  id  —  (a  +  #)]. 

We  call  it  the  tensor  of  uniform  growth  from  a-\-ib  to  c-\-id. 
The  growth  is  evidently  of  the  sort  c  -f-  id  —  {a  -f-  #)« 

A  uniform  growth  from  a  -f-  ib  to  a  number  x  -f-  iy,  if  not  of 
the  sort  c-\-id  —  {a-\-  ib),  must  be  followed  by  one  also  not  of 
that  sort  to  get  from  x-\-  iy  to  c-\-  id.  By  §  107,  the  sum  of 
the  tensors  of  the  two  growths  exceeds  the  tensor  of  a  single 
uniform  growth  from  a  -\-  ib  to  c  -\-  id.  We  say  that  the  single 
growth  is  more  direct.  A  fortiori,  the  single  growth  is  more 
direct  than  a  chain  of  uniform  growths  from  a  -f-  ib  to  c-\-id 
through  numbers  xx  +  iy, ,  x,  +  iy, ,  x%  -f  iy3 ,  .  .  .  xn  +  iyn , 
not  lying  directly  between  a  -\-  ib  and  c-\-id. 

121.  Suppose  that,  in  the  chain  of  growths  just  suggested, 

a  <  xx  <  x,  <  x%  <  .  .  .  <  ^M_2  <  xn_x  <xu<b; 
and  also 

*,  -b  ^j£^9x  ^j—y*  ^      >^-i-^-a>jf«-^-i>  *b?« 

^i_^>  ^2_^  ^  *3_*a  ^  '  '  •      xn_,-xn_2      xn-xn_,      c-xn 

By  addition  and  subtraction  of  numerators  and  denomina- 
tors we  get 

yx-b     y^b     y3-b  y«-.-b     y«-b      d-bm 

-^Z7a>  x2-a      xz-a^  "  '^  *„.,-*      xn  -  a      c  -  ay 


v^yi.y^y^yt-y^^         yn-,  -  y*    yn^il* 

Xt—  Xx        *%—  *i        **—  *r  *n-x~*x        %n  — Xx 

together  with  similar  chains  of  inequalities  beginning 


y3-y2    y*-y*    y<>-y*         yn-*—yn-*    yn - .?-, 

*3-*2'     x-x,'    xs-x4>  '  '  '  x^-x^'     xn-xn_x 


92         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

Also,  we  have 

d-  yn      d-  yn_       d-yn_  ^d  —  yd-yx      d—b 

c  —  xn      c  —  xn_i      c  —  xn_2      '  '  '      c  —  x2      c  —  x^      c  —  CL 

yn-yn-,   ^n-yn-2  ^n—yn-i  A^ll*  ^n-y*  ^n—b. 

xn     xn_i      xn     xn_2      xn     xn_^  Xn     X2      Xn     Xx      Xn     CI 

together  with  similar  chains  of  inequalities  beginning 

yn-i   —yn-2  yn-2-J>n-3  J3   ~  J2  ?2  ~  ?i 

Xn_1  %n-2  %n-2         *»-'3  "^3  %2  ■*3         *l 

In  brief,  if  we  put  for  a,  b,  c,  and  d,  respectively,  x0,  yoy 
xn  +  I,  and  yn  +  1,  while  letting  /,  k,j  be  three  of  the  numbers 
o,  I,  2,  .  .  .  n-\-  I,  with  />/,  we  have 

yk—yi    yk  - y,- 

Xk        Xt        Xk  —  Xj 

Any  number  directly  between  xk-\~iyk  and  ^i+i+^+i  & 

.  m(xk-\-  tyk)  -\-  n(xk  +  l  -f-  iyk  +  i),  where  m  and  n  are  positive  and 
m-\-n  =  I.     Therefore,  unless  £  -|-  i  =  /, 

yk-yi     myk+  nyk+i-yiyyk+i  —  ytm 

xk  —  xt      mxk  +  **k+x—*i      Vk  +  i  —  xi' 
since  the  middle  ratio  is  *»-*)+»&  +  *-*) 
In  like  manner,  unless  k  =  I  -f-  i, 

yk  —  yi+i  >  *»n +  «?»+,  —^+,  >  jWi  —  .?/+. 

■**      ^  + 1       *M#k  ~r  *^i  +  i  —  xi  + 1      *j  + 1  —  ■#*  + 1 

Then,  if  r{xt  +  *?*)  +  j(^+i  +  iyi+i)  is  any  number  directly 
between  xt  -f-  iyt  and  xl+I  -\-  iyt+xi  it  follows  that 

yk  —  {ryi  +  syl+1)      myk  +nyk  +  I  -  (ry{  +  syl+I) 
*k  —  (rxt  +  syl+l)      mxk  +  nyk  +  l  —  (rxt  +  sxl+I) 

?k  +  i  ~(ryi  +  syl+I) 
Xk  +  i-(ryt  +  syl+l)' 


GROWTHS,    RATES,    AND   AMOUNTS.  93 

Here  notice  that  as  111  approaches  unity  and  n  approaches 
zero,  the  middle  ratio  approaches  the  left-hand  ratio ;  while, 
when  ?i  approaches  unity  and  m  approaches  zero,  the  middle 
ratio  approaches  the  right-hand  ratio.  At  the  same  time, 
m{xk  +  iyk)  +  n{xkJrl  -f  iyk  +  I)  approaches,  respectively,  xk  +  iyk 
and  xk+l  +  *>*+,, 

Now  r{xt  -f-  iy!)  +  s(xl+1-\-  iyi+1)  is  any  number  whatsoever  on 
the  chain  of  growths  from  a  -\-  ib  to  c  -\-  id;  and  m(xk  -f-  tyk)  -|- 
n(*k+i  +  tyk+i)  is  any  other  number  on  that  chain. 

Imagine,  for  the  moment,  a  uniform  growth  joining  these 
two  numbers.  The  middle  ratio  above  is  that  of  the  i  to  the 
non-2  part  of  this  growth.  We  are  told  then,  by  the  inequali- 
ties, that  as  either  of  the  two  numbers  changes  its  value  along 
the  chain  of  growths  from  a  -f-  ib  to  c  +  id,  this  ratio  likewise 
changes,  growing  ever  smaller  for  a  change  of  either  number 
toward  c  -\-  id,  ever  larger  for  a  change  toward  a  -f-  ib. 

Hence  a  uniform  growth  joining  any  two  numbers  on  the 
chain  of  growths,  but  not  itself  forming  part  of  the  chain,  can- 
not contain  a  third  number  on  that  chain. 

122.  Let  there  be  a  second  chain  of  growths  joining  a  -j-  ib 
to  c  -\-  id,  of  the  same  character  as  the  one  just  treated  but 
through  zl  -f-  ivl ,  z2  -f-  iv2,  zz  -f-  iv$1  .  .  •  ,  zm -}-  ivm.  Call  the 
first  the  x  -\-iy  chain  ;  the  second,  the  z  -\-  iv  chain. 

Further,  let  every  number  on  the  z  -j-  w  chain  lie  directly 
between  a  -\-  ib  and  some  number  on  the  x  -f-  iy  chain.  In 
other  words,  let  all  the  numbers  on  the  z  -f-  iv  chain  be  of  the 
type 

r(a  +  ib)  -f  s  [m(xk  +  iyk)  +  n(xk+l  +  *>*+,)]. 

Then,  the  z  -\-  iv  chain  is  more  direct  than  the  x  -f-  iy  chain. 

To  prove  this,  we  need  to  show  that  just  as  the  growth  from 
a  -f-  ib  to  zl  -|-  ivl  will,  if  continued,  contain  some  number  on 
the  x  +  iy  chain,  so  also  will  the  growths  from  z.  -f-  ivx  to 
z2  -f-  iv2 ,  from  z2  -\-  iv2  to  z3  +  ^3 »  from  z3  +  iv3  to  z4  -\-  iv4 ,  and 
so  on,  every  growth  of  the  z  -f-  iv  chain  containing,  if  continued, 
a  number  on  the  x  -\-  iy  chain. 


94         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

Consider  a  series  of  uniform   growths   connecting  a  -\-  ib, 

*t  +^>i» *i  +tr».i  •  •  •  »  a11  to  **  +  ivk> 

Since  ^  -f-  *#4  is  on  a  uniform  growth  from  a  -\-  ib  to  some 
number  on  the  x  -|-  iy  chain,  and  likewise  zk+l  -f-  m>a+i , 

7i  —  b  =  *V~  *      **+,  —  »j 


xx—a>zk  —  a      zk  +  l  —  #A 

If  — = ,  then  zk  +  ivk  is  itself  a  number  on  the 

;r,  —  a       zk  —  a  ' 

jtr  -|-  iy  chain  ;  and  since  —  <  -^ * ,  it  follows  that 

c       zk-\-i        Zk  +  i  —  Zk 
the  growth  from  zk  +  ivk  to  *A+1  +  «>a+i  is  part  of  some  uni- 
form growth  or  other  joining  two  numbers  of  the  x  -f-  iy  chain. 

If  — >  — ,  we  may  have  x,  =zk.      If  x.  >  zk>  con- 

x1—azk  —  a  *  <  i  —    *  > 

sider  m(a  +  ib)  -\-  n(x1  -f-  y/,),   where   of  course  m  >  O,  ?z  >  o, 
and  m  -\-  n  —  I. 

Evidently  ma-\-  nx,  as  «  grows  from  o  to  I,  will  grow  from 

i                   111.    mb  +  Wi  —  b  .    Vr  —  b 
a  through  zk  to  xx ;  but  always  the  ratio . is 


ma  -f-  ^^"i  —  #     xx  —  a 
We  have  then 

vk  —  (;;z£  -|"  nyd      mb  -\-  ny,  —  b      vk  —  b 

7 : r  > : > 7 ,  if  ma  4-  nx.  >  zk ; 

zk  —  {ma  -\-  7txI)      ma  -f-  /z;^  —  a      zk  —  b 

mb  -f-  «y,  —  £      7^,  —  b      vk  —  (mb  -\-  ny,) 

but . > > , ; r,  if  ma  +  nx,  <  zk; 

ma  ~\-  nxx  —a      zk  —  a      zk  —  (ma  -\-  nxx) 

while,  in  either  case,  — -, : ^r  ~  — l . 

zk  —  (ma  -f-  nxx)  >  zk—x1 

vk  -  (mb  +  nyx)  vk  —  b 

Thus  t : r  can  take  all  values   between 

zk  —  (ma  -[-  nx,)  zk  —  a 

,  vk  —  yt  .  .    vk+I  —  vk 

and .     Among  these  values  is  — . 

*k—xl  *  **+x  —  zk 

Consequently,  if  xl^.JSk,  the  uniform    growth  connecting 
some  number  or  other  on  the  growth  from  a  -{-  ib  to  xt  -\-  iyl 


GROWTHS,    RATES,    AND   AMOUNTS.  95 

with  zk  -f-  ivk  is,  together  with  the  growth  from  zk  -f-  ivk  to 
jZk+i  +  ivk+1 ,  part  of  a  uniform  growth  joining  two  numbers  of 
the  x  -\-  iy  chain. 

But  suppose  Xj_  <  zkJ  then  — -1  <  -— -.     Repeating 

the  previous  reasoning,  with  a,  b,  x1 ,  yz ,  replaced  respectively 
by  *t*  Jx\  *2t  fat  we  find  that,  if  Xa^~  *ti  the  growth  from 
^r^  -f-  2^  to  ^h-i  +  ^2/Ar  +  I  will  yet  be  part  of  a  uniform  growth 
joining  two    numbers  on   the   x -\- iy  chain.     If   xt<ski  but 

jra  —  zk,  the  above  statement  still  holds.     Likewise  does  it  if 

3  >    * 

we  can  at  last  find,  from  among  the  numbers  xl9  xM,  x3t  .  .  .  , 

xn ,  c,  a  number  larger  than  zk .    But  certainly  c  is  larger  than  zk . 

Therefore,  always,  the  growth  from  zk  -f  ivk  to  zk+l  -\-  tvk  +  1  is 

part  of  a  uniform  growth  joining  two  numbers  on  the  x  -\-  iy 

chain. 

Since  the  z  -j-  iv  chain   is   not   identical  with   the  x  -\- iy 

chain,  at  least  one  number  determining   the   z  -\-  iv   chain  is 

different  from  any  number  determining  the  x  +  iy  chain.     Let, 

then,  zk  -\-  ivk   be   a    number   not    identical   with   any  of   the 

numbers    a  +  ib,  xx-\-iyx,    x2  +  iynf  ♦  .  .  ,  xn -j- iyn%    c-\-  id. 

vk  —  vk_x       vk+1  —  vk     1       .  . 

Because >  — — ,  the  three  numbers  zk_T-{- zvk.lt 

zk  —  zk-\      zk-\-\       zk 

*k~{-ivk,  zk+1-{-  ivk_iy  cannot  possibly  all  be  on  one  and  the 
same  growth  of  the  x  -\-  iy  chain.  Therefore  at  least  one  of 
the  two  growths  of  the  z  -f-  iv  chain  connecting  these  numbers 
is  no  part  of  a  growth  of  the  x  -\-  iy  chain.  Continue  this 
growth,  then,  till  it  meets  the  x  -\-  iy  chain,  thus  joining  two 
numbers  on  that  chain  by  a  single  uniform  growth.  A  new 
chain  is  formed,  made  up  of  these  three  parts :  the  x -\-  iy  chain 
from  a  -\-  ib  to  the  first  meeting  number,  the  uniform  growth 
from  here  to  the  second  meeting  number,  the  x  -\- iy  chain 
from  there  to  c  -\-  id.  The  new  chain  is  more  direct  than 
the  x  +  iy  chain.  Call  it  a  first  shortening  of  that  chain. 
Evidently  this  shortening  enjoys  all  the  properties  stated  as 
common  to  the  x  -\-  iy  chain  and  the  z  -\-  iv  chain.  Further- 
more, if  not  identical  with  the  z  -f-  iv  chain,  it  is  met  by  con- 


g6         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

tinuations  of  the  growths  of  that  chain  ;  otherwise  the  x  +  iy 
chain  could  not  be  met  by  these  continued  growths.  Take, 
then,  any  number,^-}-  ivl%  not  determining  the  first  shortening, 
and  continue  a  growth  of  the  z  -f-  iv  chain  from  it  to  meet  that 
shortening.  We  get  a  second  shortening.  If  this  be  not 
identical  with  the  z-\-iv  chain,  we  can  in  like  manner  get  a 
third  shortening,  and  so  on.  But  at  the  {in  -f-  i)st  shortening, 
if  not  before,  we  reproduce  the  z  -\-  iv  chain.  Thus  the  z  -f-  iv 
chain,  the  result  of  successive  shortenings  of  the  x  -f-  iy  chain, 
is  more  direct  than  that  chain. 

To  clearly  fix  in  his  mind  the  reasoning  of  this  and  the 
preceding  section,  the  student  should  draw  figures  to  represent 
the  x  -\-  iy  and  z  -\-  iv  chains,  together  with  the  typical  joining 
lines  to  which  reference  is  made.  He  will  find  that  what  is  so 
difficult  from  the  standpoint  of  pure  algebra  is  geometrically  a 
mere  truism. 

123.  Let  a  growth  of  varying  sort  connect  a  -f-  ib  with 
c  -\-  id,  and  let  z'  -f-  w^»  *"  +  W*  z'"  +  W't  be  an>T  numbers 
on  that  growth.  Furthermore,  let  the  growth  be  such  that  if 
only  z'  <  z"  <  z'",  we  shall  have 

v"  _  v>      v">  _  v" 
7  ^  „m       „/f  • 


z    —  z 

If  then  *,  +  iv1 ,  z2  +  iv2 ,  z3  +  iv% ,  .  .  .  \sm  +  Wm ,  be  num- 
bers on  the  growth,  and  a  <  z,  <  z2  <  z3<  .  .  .  <  zm_I  <  zm  <  c, 
the  chain  of  growths  gotten  by  directly  joining  these  numbers 
in  order  will  be  of  precisely  the  same  nature  as  the  z  ~\-  iv 
chain  of  the  preceding  section.  We  use  the  same  name  for  it. 
The  varying  growth  we,  in  like  manner,  call  the  z  -f-  iv  growth, 
and  use  z  -\--iv  to  denote  a  variable  number  on  that  growth. 
The  growth  is,  if  you  please,  the  result  of  (z  -f-  ^)'s  growing  in 
a  specified  way  from  a  -\-.ib  to  c  -\-  id. 

The  numbers  a  +  ib,  z1  -\-  iv, ,  z2  +  iv21  .  .  .  ,  zm  +  tvm9 
c-\-  id,  are  the  only  numbers  common  to  growth  and  chain. 
For  that  the  number  m(xk-{- iyk) -\- ?i(xk+l-{- iyk+l),  on  the 
chain,  should  be  also  on  the  growth  would  contradict  the 
inequalities  used  to  define  the  character  of  the  growth. 


GROWTHS,   RATES,   AND  AMOUNTS.  97 

Suppose  z  -f-  iv  to  grow  from  a  +  ib  to  xk  -f-  /)>*,  and  then 

on  to  ^  -j-  z#.     At  the  same  time,  the  ratio grows  from 

to  — -,  and  then  on  to  -.     The,  by  itself,  mean- 


&  —  Zk       zk  —  Zk  c  —  zk 

ingless  expression  is  hemmed  in  by  ratios  that  differ 

Zk  —  zk 

as  little  as  we  please,  and  so  becomes  a  symbol  of  the  limit  of 
these  ratios. 

By  §  1 1 8,  there  is  a  number,  call  it  xk-\-  iyk,  such  that 

yk  —  vk  _  Vk  —  vk     and    yk  —  n-j  _  vk-i  —  Vk-x 
xk  —  zk       zk       zk  xk       zk_x       zk_1  —  zk_1 

Thus  are  defined  a  series  of  numbers  x1-{-iyl,  x2-\-iy2, 
x3-\-iy2>,  .  .  .  ,  xm+t  +  iym+ 1  j  forming,  with  a  -f-  ib  and  c  +  id, 
the  basis  of  an  x  -\-  iy  chain.  The  only  numbers  common  to 
it  and   the  z  -\-  iv  growth  are  the  numbers,  a  -\-  ib,  xl-\-iyli 

x2  +  iy2 »  •  •  •  »  xm  +  ij'm  >  c  +  M*  a^so  on  tne  z  -f  iv  chain. 

The  present  x  -f-  (y  and  ^-)-^  chains  are  related  as  were 
those  of  the  preceding  section  ;  i.e.,  any  number  on  the  z  -\-  iv 
chain  lies  between  a  -\-  ib  and  some  number  on  the  x-\-iy  chain. 
We  leave  the  proof  to  the  student.  The  proof  established,  it 
follows  that  the  x  +  iy  chain  is  less  direct  than  the  z -\- iv 
chain. 

124.  Take  on  the  z  -{-iv  growth  between  each  pair  of  num- 
bers zk  -{-  ivky  zk+1  -f-  ivk+I ,  a  number  zkf  +  ivk'}  and  then  through 
a  +  ib,  zQ'  +  ivj,  z,  +  iv, ,  z/  +  iv/,  z2  +  iv2 ,  .  .  .  ,  zj  +  ivm', 
c  -\-  id,  form  a  zf  -f-  *V  chain.  From  this  chain  form  an  or'  +  */ 
chain,  as  the  x  -{-  iy  chain  was  formed  from  the  z  +  iv  chain. 
Evidently  the  x  -f-  iy  chain  is  less  direct  than  the  x'  -f-  ?j/' 
chain,  that  less  direct  than  the  z'  -j-  iv'  chain,  and  the  z'  -f-  iv' 
chain  less  direct  than  the  z  -\-  iv  chain. 

In  the  same  way  we  can  go  on  forever  getting  z"  +  iv" , 
z'"  +  iv'" ,  .  .  .  ,  z{n)  -(-  ^(M)  chains,  each  less  direct  than  the 
preceding  one ;  and  at  the  same  time,  x"  +  iy",  x'"  -\-  iy'" , 
.  .  .  ,  x{n)  +  iy{H)  chains,  each  more  direct  than  the  preceding 


98         AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

one.  But  always  the  z(n)  -f-  iifn)  chain  is  more  direct  than  the 
x{n)  +  iy[n)  chain. 

In  other  words,  the  sum  of  the  tensors  of  the  growths  in 
the  x(n)  -\-  iy{n)  chain  approaches  an  inferior  limit  that  either 
■equals  or  exceeds  a  superior  limit  approached  by  the  sum  of 
the  tensors  of  the  growths  in  the  z{n)  +  iv[n)  chain. 

We  can  prove  the  two  limits  equal. 

For  convenience,  call  any  two  successive  numbers  common 
to  the  z  +  iv  growth  and  the  z{n)  +  iv{n)  chain,  a  -\-  ib  and 
a'  +  ib'. 

b-b      b'-b      b'-b'        '         t 

Suppose  a  >  a,  then   >  — >  — 7 .      r  or  the 

rr  a  —  a      a  —  a      a  —  a 

b-b  h'  -b'  q        ,   q'        .  . 

ratios and  —, -.  we  can  put  -  and  -, ,  with  p2  -\-  (f 

a  —  a  a'  —  a  P  P  * 

=/' 2  +  4 2  —  1  and/'  >p>0.    Let  a'  —  a  =  h,  and  ^  —  ^-,  =  £. 

The  number  of  the  x{n)  -f-  iy{n)  chain  determined  by  the 
numbers  a  +  *#  and  a'  +  *#'  of  the  z{n)  -f-  ^(M)  chain  is  simply 
that  number  which  can  be  reached  both  by  a  growth  of  the 
sort  p  +  iq  from  #  -f-  ib,  and  a  growth  of  the  sort  p'  -f-  a?1  from 
#'  +  ^'-     Using  the  notation  of  §  118,  the  number  is,  in  fact, 


a  +  mp  -\-t{p-\-  mq)  —  a'  -f-  ^'/>'  -f-  i(p'  +  *»V). 

By  §  120,  m  and  w'  are  numerically  equal  to  the  tensors  of 
the  growths  from  a  -f-  ib  and  #'  -f-  ib' .  Call  these  tensors  /  and 
t'  respectively.     Then 


p\V-b)-g'{a'-d) 
p'q-q'P 

and  ,=  &-*-**-*>._ 

P'q  -  q'p 


V 

-b 

qf 

h 

a' 

—  a 

P' 

p 

<2 

P 

k 
V  • 

-b 

h 

a'  - 

—  a 

p' 

k 

' 

GROWTHS,   RATES,   AND  AMOUNTS.  99 

The  uniform  growth  from  a  -f-  ib  to  a'  -\-  ib'  and  forming 
part  of  the  z{n)  -\-  iv{H)  chain  has  the  tensor 


+  V{a'  -  jf  +  {bf  -  Sf  =  h. a/7+  (£—£  =  I, 


say. 


jf  1  r 

Because  of  the  inequalities  —  >  —, >  —7,  we  have 

pa  —  a      p 

h       ,     h' 
p  p 

Thus  /  lies  between  numbers  differing  by  less  than 


Now      ?-7=(?-^)^G+i?)• 

But 

1  _  _!  =  P'2-f  =  (f  +  W-W  +  f*)?  -t_C 
f    p'*       p*p"  fp"  p>    p'" 

and  h  +  l>i+Z> 

P  P  P  P 

"  P  '  / •'     V     /v  '  V     /'  ~~  /     /  ~"   ' 

and  the  /  inclusives  differ  by  less  than  kk. 

Observe  that  the  last  factors  in  the  values  of  /  and  /',  above, 
are  less  than  unity,  while  the  sum  of  the  numerators  of  these 
factors  is  k.     It  follows  at  once  that 

l(b'-b      q\  ..      ,         .        __ 

/  =  -A—. —, ]  —  something  less  than  kk, 

k\a  —  a     p'}  b 

t'  =  -\±- 7-—~j  +  something  less  than  kk, 

and  t  -f- 1'  =  /-[-  something  less  than  M. 


IOO      AN  INTRODUCTION    TO    THE  LOGIC   OF  ALGEBRA. 

As  formerly,  suppose  a  -J-  ib  and  c  -\-  id  to  be  the  terminal 
numbers  of  the  z  -\-  iv  growth.  The  x{n)  +  iy[n)  chain  is  made 
up  of  growths  having  the  tensors  *,,  //,  t2,  t/,  t3,  t3  ,  .  .  .  The 
z{n)  _j_  iv(n)  chain  is  macie  Up  0f  growths  having  the  tensors 
/,,  /a,  /3,  .  .  .  By  what  we  have  just  shown, 

ti  +  tZ-KkA,  t2  +  t2'  -l2<k2h2,  t3  +  t3f  -l3<k3k3,  ... 

Therefore  the  two  chains  differ  by  less  than 

kA  +  KK  +  k3h3  +  .  .  .   =  K{c  -  a), 

where  iTis  the  largest  of  the  numbers  kIt  k2,  k3,  .  .  . 

Now  c  —  a  is  a  fixed  number,  and  iT  can  be  made  as  small 
as  you  please.  Consequently  K{c  —  d)  is  as  small  as  you 
please,  and  the  x[n)  -\-  iy{n)  limit  is  identical  with  the  z{n)  -f-  w{n} 
limit.     We  call  this  limit  the  amount  of  the  z  -f-  iv  growth. 

It  may  be  objected  that  by  different  distributions  of  the 
numbers  on  the  varying  growth  we  could  get  different  final 
limits.  This,  however,  is  easily  shown  to  be  impossible.  For 
if  Z'  and  X'  are  the  amounts  of  the  less  and  more  direct  chains 
determined  by  any  distribution,  and  Z"  and  X"  the  correspond- 
ing amounts  for  any  other  distribution,  while  Z  and  Jf  are  the 
amounts  determined  by  taking  for  our  z  -\-  iy  and  x  -\-iy  num- 
bers all  the  numbers  contained  in  either  and  each  distribution, 
we  shall  have 

Z'<Z<X<Xf,     andalso     Z"  <  Z  <  X  <  X" . 

Consequently  the  limit  approached  by  increasing  the  number 
of  numbers  in  either  distribution  is  identical  with  that  ap- 
proached by  increasing  the  number  of  numbers  in  the  combined 
distribution. 

Of  course,  instead  of  showing  that  the  difference  of  the 
two  limits  must  be  less  than  K{c  —  a),  we  could  have  shown 

that  it  was  less  than  f  — — — )^^  where  H  is  the  largest 

of  the  numbers  hxi  h2>  K,  .  .  . 


LOGARITHMIC  GRO  WTHS  AND  DOUBLE-NUMBER  PO  WERS.  IOI 
125.   No  essential  difference  in  treatment  is  necessary  for  a 

y  —  y 

varying  growth  in  which  the  ratio continually  increases 

with  increase  of  x.  Observe  also  that,  if  the  growth  is  a  non- 
varying  one,  the  method  is  still  applicable,  but  superfluous ; 
since,  in  that  case,  the  amount  of  growth  does  not  differ  from 
the  tensor  of  the  growth. 

All  growths  whatsoever  can  be  broken  up  into  parts  for 

y  —  y 

which  either increases  with  increase  of  x,  or  decreases 

x  —  x 

with    increase    of   x,  or   remains    constant.     The  sum   of   the 

amounts  of  growth  of  the  parts  is  the  total  amount  of  growth. 

This  entire  investigation  could  have  been  carried  through 

referring  all  the  numbers  to  a/  -f-  iq  instead  of  a  standard  system, 

y  —  y 

and   in  place  of ,  the  ratio  of  the  vanishing  differences 

of  the  i  and  non-z  parts  of  consecutive  numbers  on  the  varying 
growth,  substituting  the  ratio  of  the  vanishing  differences  of 
the  ip  —  q  and/  -f~  iq  parts  of  those  numbers. 


V.    Logarithmic  Growths  and  Double-number  Powers. 

126.  We  have  seen  (§  113)  how  (p  -f-  iq)H,  by  the  growth  of 
11  from  zero,  could  become  any  complex  unit  whatever :  how, 
indeed,  passing  once  and  only  once  through  each  and  every 
complex  unit,  it  could  grow  to  (/  -f-  iqf  =  1,  the  starting  value. 
What  is  the  total  amount  of  growth? 

Consider  that  portion  of  the  growth  in  which  both  the  i  and 
non-z'  parts  of  the  growing  number  are  positive.  That  is  to 
say,  let  x  -\-  iy,  keeping  x2  -\-  y2  =  1,  grow  from  o  -f-  i  to  1  -\-oi. 

y  —  y 

As  we  shall   presently  see,   the   growth    is  such   that 

decreases  with  increasing  x.  For  convenience  in  calculation, 
we  so  take  the  numbers  xl  +  iyiy  x2-\-  iy2,  .  .  .  ,  that  the  ten- 
sor of  the  difference  of  each  two  adjacent  numbers  shall  be  the 
same.  For  example,  if  between  i  and  1  we  take  only  the  num- 
ber  xl-\-iyl1  this  shall  be  i*  =  \/\  -\-  i  \f\\  if  we   take  three 


102       AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

numbers,  these  shall  be  x%  -j-  iy1  =  *'*,  x2  -\-  iy2  =  O,  ^3  -f-  *X  —  **• 
Of  course  we  always  take  doubly-positive  roots,  so  that  these 
last  numbers  are 


£(</2-  V2  +  W2+  V2)>4(  4/2+*  4/2),i(v/2+  4/2-f  z-y/2-  4/2). 

After  this  fashion,  interpolate  1023  numbers.     They  will  be 

1024  -  k 

xi  +  iy*  —  ****i  ^2  +  iya  —  ******  •  ••>•#*+ *?*  ==  ^'  I024  ,  •  •  •  > 

The  student  may  show  that  ^  -f-  y^  =  yl02A-k~\- ^1024- *>  so 
that  *I023  +  z>1023  =  j^  +  Mr, . 

We  have 

(1024  -  k  - 1         1024  -  k\ 
i   "T   -  *  *"*  J 

=  r(i  -  ***«  ~  f[i  -  u  +  ^)]  =  +  V2(i  -7,). 


Thus,  the  amount  of  growth  is  1024  4/2(1  —  j/x). 

Let  the  student  show  that  y512  =  £  4/2,  jj>256  =  -J  V2  -f-  4/2, 

^28  =  iV^  +4/2"+  4/2,  .  •  .  ,  >*  =p  i  4/2  +2j2,,  and  finally 
that 


Vr-H  \\2V 


2+  \   2+ 


f[2+f  ^+f[^+f[^+l|(^+^+f 


"// 


In  actually  performing  the  calculation  we  begin  at  the  right 
and  proceed  toward  the  left.  If  a  sufficient  number  of  decimal 
places  be  used,  it  will  be  found  that 

yx  =  0.999998823449,  xl  =  0.0015339802, 


4/2(1  —  y,)  =  0.0015339806,      1024  4/2(i  —y,)  =  1. 5 70796 1. 


The  .number  of  which  1024  4/2(1  —  yT)  is  an  approximate  value 

is  symbolized  by  -,  so  that  n  is  very  nearly  3. 141 5922. 

To  find  the  error  in  the  value  of  n  we  calculate,  by  the 
method  of  §  124,  the  error  in  taking  for  the  growth  from  i  to 


LOGARITHMIC  GRO  WTHS  AND  DOUBLE-NUMBER  PO  WERS.  IO3 

**i  +  W  the  tensor  of  that  growth.     This  multiplied  by  1024  is 
the  total  error  in  the  growth  from  /to  1.     We  have 

y  —  ji  _  y2  —  y?  x  +  xx  _      %  +  *i  >     **        > 

^^  ""  *2  -  *rj+~y>  "  ~7T^  <  "J/    *x  <*  >  °* 

1/  —  y\      y\  —  y-i  y  v 

But  = ,  if  x  —  xY ;    and  consequently  -- — 

x        x\         %i        %\  xx  —  xx 


o 

—  -  =  o,  and 


Xj 

zzz  —  — 

y* 

Likewise,  = , 

x  —  x            y 

*o  —  *o                 J^c 

^1024     y  1024 * 

"^1024           •^1024                    O 

(Thus,  as  x  grows  from  o  to  1,  - decreases  fromoto— 00 .) 

X         X 

If  for  a,  b,  a',  b\  of  §  124,  we  write  x0,yoy  xlfylf  the  /  and 
/'  of  §  124  become 


I  £,  J>x  £, 

I  —  i/T  2(1  —  y.) 

.-.  /  =  t'  = —     and     /  +  /'  =  — —  =  0.0015339810. 

x i  x^ 

The  error  is  thus  less  than  0.0CO0000004  for  the  growth 
from  i  to  xx  +  iyt »  and  less  than  0.0000005  for  the  growth  from 
z  to  1.  Therefore  the  amount  of  growth  certainly  lies  between 
1. 5707961  and  1.5707966.     Consequently 

3.1415922  <  ;r  <3.i4i5932. 

More  accurate  calculation  gives  n  =  3. 141 59265  .... 

When  unity  grows  through  all  complex  units  around  to 
unity  again,  the  total  amount  of  growth  is  made  up  of  the 
amounts  of  growth  from  1  to  t,  from  i  to  —  1,  from  —  1  to  —  i, 


104      AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

and  from  —  i  to  unity.     Each  of  these  growths  has  the  same 

amount  as  that  from  i  to  i.     The  total  amount  of  growth  is 

therefore  2n. 

127.  The  expression  in  when  n  starts  to  grow  from  zero, 

does  itself  start  to  grow  from   unity.     What   is  the  rate   of 

growth   of  in  compared  to  »?     When  n  grows  from  o  to  ±  1, 

i"  grows  from  1  to  ±  i ';  and,  moreover,  if  during  the  growths  n 

takes  a  value  between  n'  and  n" ,  in  at  the  same  time  takes  a 

i°  —  i°, 
value  between  in'  and  in".     Consequently ,  the  symbol 

for  the  rate  of  growth  of  in  compared  to  n  when  n  =  o,  lies 

ih  —  I  i~h  —  l      ,  =  _ 

between ; —  and -7 —  where  1  ^  h  >  o. 

h  —  h  ^ 

Put  h  =  I,  -g-,  4,  -g-,   .   .   .    i 0 2 4 . 

Then  the  rate is  approached,  on  the   one  hand,  by 

i  -  1,  2(i*  -  1),  40'*  -  1),  8(#*  -  1),  .  .  .  ,  io24(/t*«  —  1) ; 
on  the  other,  by 

\—i-\2{\—i~  *),4(i  —  ^ -*),  8(1  —  #-*),  .  .  .  ,  1024(1  —  *^). 

Now  1024  (**«fr«  —  1)  =  1024(7,+  &r,  —  1),  where  xx  andjj\ 
have  the  meaning  given  them  in  §  126. 
But 

1024(7,  +  ixx  -  1)  =  -  10244/ *—^  •  ^2(1  —7,)  +  1024^ .  t 


2         '        2' 


n 


since  1024  y  2(1  —  y^)  and  1024^,  are  each,  very  nearly,  -. 

2 

In  the  same  way 

1024(1  —  &**)  =  -y  — — 1  + 1-  ,  very  nearly. 


LOGARITHMIC  GRO  WTHS  AND  DOUBLE-NUMBER  PO  WERS.  I05 

Were  we  to  take  //  smaller  and  smaller,  the  expressions  for 

which  we  have  written  -  would  become  more  and  more  nearly 

-,  while  y  —  would  become  more  and  more  nearly  zero, 

both  approachings  going  on  without  limit.     Thus,  the  non-z 

i°  —  i° 
part  of cannot  be  ever  so  little  either  positive  or  nega- 
tive, while  the  i  part  cannot  differ  from  -.     The  growth   rate, 

then,  of  in  compared  to  n  when  n  =  o  is  — . 

in  —  i" 
In  the  same  way  — ,  the  general  rate  of  growth  of  in 

compared  to  n>  is  as  near  as  one  pleases  . ,  when  h  ap- 

proaches  zero. 

in  +  k_jk  ih—i        .      in 

But  j— -, =  in 7—  as  in  •  — . 

n  -\-  h  —  n  It  2 

Thus  the   rate  of  growth  divided  by  growing  number  is 

in 
constant  and  equal  to  — .     Compare  now  §§  92,  93. 

The  result  just  reached  may  be  stated: 

By  the  powering  of  i,  unity  or  i°  grows  at  the  logarithmic 

in 
rate  —— . 
2 

n 
But  i1  =  i,  and  the   amount    of  growth  from   unity   is  -• 

Likewise  the  amount  of  growth  from  unity  to  *"  is  — .   Hence 

for  unity  to  grow  at  —  logarithmic  rate  means  to  grow  keeping 

tenser  constant  and  the  amount  of  growth  from  unity  always  - 


times  the  power-index. 


106      AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

Similarly,  for  unity  to  grow  at  logarithmic  rate  i  means  to 
grow  keeping  tensor  constant  and  the  amount  of  growth  fro7n 
unity  equal  to  the  index  of  the  power. 

Notice  that  as  the  growths  start  from  unity  their  sorts  are 
z,  while  in  passing  any  complex  unit  p-\-  iq  their  sorts  are 
ip  —  q  =-.  i  (p  -J-  iq):  the  sort  of  growth  divided  by  growing 
number  is  always  i.  We  say  that  the  growths  are  of  the 
logarithmic  sort  i. 

The  number  by  whose  powering  unity  grows  at  the  loga- 

rithmic  rate  t  must  be  the  number  whose  —  th  power  is  i\  for  i 

can  be  reached  both  by  unity's  growth  at  logarithmic  rate  — 
with  regard  to  zero  growing  to    i,   or  by  unity's  growth  at 
logarithmic  rate  i  with  regard  to  zero  growing  to  -  .  Thus,  the 

2 

base  for  logarithmic  rate  i  is  in. 

128.  Furthermore,  by  analogy,  i  can  be  reached  by  unity  s 

growth  at  logarithmic  rate  unity  with  regard  to  zero  growing 

in 
t-ward  to  — . 
2 

This  definition  gives  for  the  base  by  whose  powering  unity 

2  /   2  \iir 

grows  at  logarithmic  rate  unity  i2n ,  since  1**7  ■  ==  i.    But  other- 
wise this  base  is  e  =  2.71828  .... 

—   21  2  IT 

Consequently    i9   =  e,     e*=i1T,     iz  =  e   2; 

■zk  far  2kg  ir 

eki  =  i-t     iki  =  e~*,     (e*i?)gi  =  f^~.e~£*; 

{ew)         =  i  .1  - 

Now  e^i1  is  any  double  number  whatsoever,  and  so  is/-|-^ 
We  have  therefore  shown  that  any  double-number  power  of  a 
double  number  is  a  double  number.     E.g., 


(i+0,  +  '=|_V2-(-^'  +  -j75)J    =.(*T.^+'sB#-.»*. 


j0.73o 


LOGARITHMIC  GRO  WTIIS  AND  DOUBLE-NUMBER  PO  WERS.  107 

Since  e°  =  1  and  ^°-693  =  2,  e'0-43*,  the  tensor  of  the  number, 

lies  between  £  and  1.  Because  i°n°  =  W  ,  the  amount  of 
growth  from  unity  to  get  the  number  is  1.147.  On  the  diagram 
of  ?J  114  this  result  would  be  represented  by  a  point  rather  more 
than  £  from  the  nul-point  on  a  line  from  the  origin  passing 
about  midway  between  9  and  10,  O.73  of  the  way  from  I  to  £ 
on  the  growth  through  1,  2,  3,  .  .  . 

In    like    manner,    calculate    and    plot   (1  — if '"  \  (-§-  -f-  i$)in> 
(4-3/)1-1,  (-  i)1',  I"1',  I<%  (-*)-'. 

129.  In  the  last  section  we  saw  that  unity  growing  at  loga- 

2 
rithmic  rate  i  with  regard  to  zero  growing  to  1,  became  i*. 
Had  it  grown  at  the  uniform  rate  i,  it  would  have  become 
1  -j-  z.  Similarly,  had  it  grown  uniformly  at  rate  i  with  regard 
to  zero  growing  to  $-,  and  then  at  the  uniform  rate  z(i  +  \i) 
with  regard  to  the  further  growth  of  \  to  1,  it  would  have  be- 
come 1  +  \i '  -f-  \i(\  +  \i)  =  1 1  -| J  .     In  like  manner,  if  for 

,      r                  123                        n— I 
the  growths  from  o  to  -to -to-  to  .  .  .  to to    1,   unity 

grows  at  the  successive   uniform   rates  I,  i\\  -| — ),  i\\  -\ — J  > 

If  «  =  1  000000,  the  tensor  of  1  A —  is  i/i.oooooooooooi, 
and  the  tensor  of  1 1  -| — )    is 

a  000  oon/ 
I.OOOOOOOOOOOI500000  =         A  /  I. OOO  OOO  OOO  OO  I1000000000000. 

Thus  r(i-f-)    is  very  closely  the  2000000th  root  of  e.     If 

we  put  n  =  1  000000000000,  T\\  -| — J  is  still  more  closely 
the  2  000  000 000  oooth  root  of  e.     Increasing n  without  limit  will 


108       AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

increase  without  limit  the  index  of  the  root  we  must  take  of  e 

(  i\n 

to  get  71  I  +  —  J  •     Consequently,  by  making  n  large  enough, 

r(i  +  i),    7(1  +  -9',    r(l  +  J)'  ,  .  .  .  ,  7(l  +  i)"   are    all 

unity. 

The  successive  growths  by  which  we  have  supposed   unity 

/  i\n  .  I 

to  become  (  1  — I ]    have  each  for  its  tensor  — th  of  the  tensor 

V      *    »/  n 

of  the  number  from  which  it  is  supposed  to  grow.  Therefore 
the  sum  of  the  tensors  of  all  these  n  growths  is  unity. 

Since  the  chain  of  growths  is  through  numbers  on  a  vary- 
ing growth  from  unity  of  logarithmic  sort  i,  and  since  the  ten- 
sors of  the  growths  are  as  small  as  you  please,  the  sum  of  the 
growths   is  as  near  as  you  please  the  amount  of  the  varying 

growth  from  unity  to  ( 1  -| —  J  .  But  when  a  growth  is  of  loga- 
rithmic sort  i  and  the  amount  of  growth  from  unity  is  unity, 
the  number  reached  by  the  growth  is  z2  =  e\  Now  e  is 
fi-| — )    and**'is(i-| — J    .     Therefore 

(■ +-0"= (■+=)"• 

Also,  of  course, 

MM.4r=(-+r-('-r=(°+3" 

provided  always  that  n  is  taken  large  enough. 

130.  Consider  now  the  expression  (i-| J  .     Just  as 

the  numbers  ( 1  -| — ),  ( 1  -| — j  ,  ( 1  -] — ),...,  are  on  a  growth 
of  logarithmic  sort  i,  so  we  should  naturally  look  for  the  num- 
bers    1  + ~,    1 1  +  - —  )  ,    I  1  +  - — -    )  ,  .  .  .  ,    on    a 


LOGARITHMIC  GROWTHS  AND  DOUBLE-NUMBER  POWERS.  ICX) 

growth   of    logarithmic    sort  /  -j-  qit  i.e.    on  a  growth   always 

(/  +  #*)'warc*  ^rom  tne  growing  number.     By  assigning  a  proper 

k 
value  to  k  any  number  on  such  a  growth  is  /*  >    «.     Our  ex- 
pectation will  then  be  justified,  if  we  can  show  that  f  1  -j--  ~*  "  J 

_  ^it+in-^     This  is  easy.     For 

+  —Jz—J  =V  +  nJ    \l  H n ) 


k  k  qi  k 


=  e>1  p  .  en  ' ■  +>  +  n  —  ™[pJrqi) 


since  -    ■    .  _r~  is  nearer  than  anything  to  qi  when  n  is  large. 

Notice  also  that  T  (1  +^±i?)*=  <i>  =  (r  +^*. 

To  get  the  amount  of  growth  of  p  -f-  £f  logarithmic  sort 

from  1  to  ^+<7'  ,  we  add  together  the  tensors  of  the  successive 

growths  from  power  to  power  of   1  -\ till  (1  -|- — ) 

is  reached. 

.     ^p+qi       1 

The  1st  tensor  is  T =  - ; 

n  n 

-  2d    -    -  T(!^.t±si\  =  ^.i, 

\  n      I  n 

«    3d       -      "  T[{e^\L±^=Ml, 

The  kth  tensor  is  T  H/^)*"1.  Ltli\  =  i?'*  -. 
Consequently  the  sum  is  -(  1  -j-  e*  -f-  eH  -j-  .  .  .  -f-  e   *    j. 


IIO      AN  INTRODUCTION    TO    THE   LOGIC   OF  ALGEBRA. 

At  the  same  time  that  these  growths  are  taking  place  the 
tensor  of  the  growing  number  has  the  growths 


p     p  i       p2J  p  tzSit 

n     n         n  n 

kp 
But  the  first  value  of  the  tensor  is  I  and  the  last  en,  while 

its  growth  is  of  a  non-z  sort.     Hence 

kp  pi  P_  2_P  (k-l)p\ 

en  —  I  =  -I  i  +  en-\-  en  -j-  .  .  .  -\-  e    *    1. 

Comparing  this  sum  of  the  growths  of  the  tensor  with  the 
sum  of  the  tensors  of  the  growths  above,  we  have  for  that  sum 

I  /  k2 

p\en-1 

When/  =  i  and  so  the  growth  is  of  a  non-2  sort,  this  is 

*                                                                   i 
£n  —  I  ;  while  for  p  = —  i,  it  is  I 5-:    results  which  verify 

•our  formula.     Again,  if  /  =  o,  and  the  growth  is  of  a  pure  i 

I  ~\° 

sort,  the  amount  of  growth  is  ^— =  -,    another    verifica- 

o  11 

tion. 

Observe  that  whether  the  logarithmic  rate  be  p  -f-  qi  or  some 
non-/  multiple  of  p  -\-  qi  in  nowise  affects  the  results.  The 
amount  of  growth  depends  solely  upon  the  logarithmic  sort 
and  the  starting  and  terminal  numbers  of  the  growth. 

Notice  some  geometric  interpretations.  The  circle  is  the 
result  of  a  growth  of  logarithmic  sort  i ;  the  spiral  drawn  full, 
of  a  growth  of  logarithmic  sort  I  -\-z;  the  spiral  drawn  broken, 
of  a  growth  of  logarithmic  sort  1  -|-  (27T  -f-  \)i.  Several  points 
on  each  growth  are  numbered.  The  student  should  satisfy 
himself  that  the  numbering  is  correct. 

The  direction  of  the  spiral  growths  at  the  start  from  unity 
is  indicated  by  arrows.  Notice  that  each  spiral  cuts  all  rays  at 
the   same   angle  that   it   there  cuts   the    unit   ray.     The    one 


LOGARITHMIC  GRO  WTHS  AND  DOUBLE-NUMBER  PO IVERS.  1 1 1 


growth  is  always  (i  -\-  z)-ward,  the  other  [i  -f-  (27t  +  i)z]-ward 
as  to  the  ray  from  which  it  momentarily  grows. 


l+f    B-(27r+l")» 

—e 


fcite 


Hix 


To  the  left  of  the  figure  is  show  a   construction   for  the 
amount  of  growth.     BE  is  the  amount  of  growth  on  the  broken 


112       AN  INTRODUCTION    TO    THE   LOGIC   OF  ALGEBRA. 

spiral  from  5to  I,  DC  the  amount  of  growth  from  I  to  C,  while 

AF  is  the  amount  of  growth  on  the   unbroken  spiral  from  A 

to  i. 

To  prove  the  construction   right  take   the  last  case.     We 

i  -  e  ~  »       OG  —  OA 

have  amount  of  growth  = =  -.- = ——  sr  AF. 

&  I  -j-   |/2        ^£-^  y4F 

131.  We  have  represented  ^I  +  *  as  reached  by  two  growths; 

one  of  the  1  -f-  i,  the  other  of  the  I  -f-  {271  -j-l)*sort.     Obviously, 

it  is  reached  by  all  growths  of  the  sorts  1  +  {znn  +  \)i  gotten 

by  giving  to  n  all  integral  values  ;  for  ^*+(»**+*)*  —  ^i  +  f^a™> 

=  **  +  ',  iM  =  ez  +  {.      The   1  -f-  (2//7T  +  1)  spiral  is  such   as  to 

make  n  turns  and  — th  of  a  turn  while  the  tensor  of  the  grow- 

27T  fa 

ing  number  grows  from  1  to  e. 

Suppose  we  wish  the  k\\\  root  of  *»+(«**+ 0«,     By  our  or- 

1        i         2«   . 

dinary  rules,  this  should  be  ek ,ek .  ek  .  The  first  factor  is  the 
tensor  of  the  root  and  independent    of  n.     The   last   can  be 

written  e  kl  ,  where  k'  is  the  remainder  after  dividing  71  by  k* 
Since  k'  can  be  any  one  of  the  numbers  o,  I,"  2, .3,  -.  .  .,  k  —  lt 

according  to  the  value  assigned  to  n,  we  thus  get  k  and  no 
more  distinct  values  for  the  second  factor.  In  other  words, 
every  root  of  eI  +  {2n  +  l)i  can  be  reached  by  less  than  a  single 
turn  of  some  logarithmic  growth  from  unity,  and  every  /£th 
root  is  on  some  one  of  the  k  growths  of  the  logarithmic  sorts 

1  +  2,1  +  (27T  +  i>;  1  -f  (47r  +  i>; . . .,  1  +  (2 . k—  1 .  71+  \)i. 

If  the  growth  be  specified  the  root  is  uniquely  determined. 
Thus  : 


Now  any  double  number  is  ea  +  bi,  where  b  <  in.     The  k 
distinct  /£th  roots  of  this  double  number  all  have  the  tensor 

a 

ek,  and  lie  on  the   k  growths  of  the  logarithmic  sorts  a  -j-  bi> 
a  +  (27t  -\-  b)i,  a  +  (47V  -f-  b)i',  .  .  .,  a  +  (2  .  k  —  1  .  n  -f  b)L 


LOGARITHMIC  GA'O  WTHS  AND  DOUBLE-NUMBER  PO  WERS.  1 1 3 

Consequently  each  can  be  reached  by  less  than  a  single  turn  of 
a  logarithmic  growth  from  unity. 

Similar  remarks  apply  to  raising  to  fractional  powers,  the 
denominator  of  the  fraction  giving  the  number  of  roots.  As 
we  increase  the  denominator  of  the  fraction,  the  number  of 
roots  forever  increases  and  they  approach  closer  and  closer 
together.  Remember  now  that  we  hem  in  incommensurables 
by  fractions,  only  by  indefinitely  increasing  the  denominators 
of  the  fractions.  It  follows  that  an  incommensurable  power 
of  a  number,  the  growth  on  which  it  lies  not  being  specified,, 
is  as  near  as  one  pleases,  any  number  having  the  right  tensor. 

132.  To  say  that  any  number  c  -f-  id  can  be  reached  from 
unity  by  an  infinity  of  distinct  logarithmic  growths  is  the  same 
as  saying  that  every  number  has  an  infinity  of  logarithms 
to  base  e\  for  the  logarithmic  rate  that  determines  the  growth 
is  nothing  else  than  a  logarithm  of  the  number  to  the  base 
e.  In  fact,  if  a  +  ib  is  a  logarithm  of  c  +  id,  so  also  are 
a  -f-  (27t  -j-  b)i,  a  -f-  (4^  +  b)i,  .  .  .,  a  -f-  (2nn  -f-  b)i'.  That 
logarithm  whose  i  part  is  less  than  2n  but  not  less  than  zero  we 
call  the  logarithm  of  the  number.  On  the  growth  determined 
by  it  lies  that  £th  root  of  the  number  whose  sort  is  between 

I  and  e  k  .  Unless  otherwise  specified,  we  take  any  fractional 
or  incommensurable  power  of  the  number  upon  this  same 
growth. 

133.  Usually,  log,  {c  -j-  id)  and  log,  (c'  -\-  id')  are  numbers 
of  different  sorts,  and  so  a  logarithmic  growth  from  unity  con- 
taining one  of  them  will  not  contain  the  other.  We  can,  how- 
ever, in  general  find  three  numbers  &,  p,  q,  (p2  -f-  g*  =  1)  such 
that  the  growth  from  ekt  of  a  logarithmic  sort  /  -f-  iq  shall  con- 
tain both  c  +  id  and  c'  -\-  id' .  For,  suppose  log,  (c  -J-  id)  = 
a  +  ib  and  log,  {c'  +  id')  =  a'  -\-  ib' .  Then  c'  +  id'  = 
ga'-a  +  iv-  b  ,(c-^-id)f  and  the  wished-for  growth  is  of  the 
sort  a!  —  a  +  i  (br  —  b).     This  determines/  and  q.     From 

q 
we  have  g  =  a  -r-  A  and  thence  k  =  b  —  gq  z=  b  —  a-. 

P 


114      AN  INTRODUCTION    TO    THE  LOGIC  OF  ALGEBRA. 

Calculate  k,  p,  and  q  for  c  -J-  id  =  el  +  i  and  d  +  id'  =  e2  +  3% 
drawing  a  diagram  to  show  the  results.  Show  that  the  method 
fails  when  and  only  when  c2  -f-  d2  =  <;'2  -j-  <^/2,  and  that  then 
both  numbers  lie  on  a  growth  of  logarithmic  sort  i  from 
Yc*  +  d*. 

134.  Suppose  we  wish  the  (d  -f-  z^')th  power  of  £  -|-  *#,  and 
that  c-\-id=  ea  +  ib.  The  desired  power  is  e^a  +  *B  <*'+  «')f  and  is 
the  result  accordingly  either  of  unity's  growing  at  the  logarithmic 
rate  d  -f-  *</'  with  regard  to  zero  growing  (0  -f"  z#)-ward  to 
<2  -|-  ib,  or  of  unity's  growing  at  logarithmic  rate  a  -\-  ib  with 
regard  to  zero  growing  (d  +  zV/')-ward  to  d  +  ftf'.    The  growth 

makes  the  same  angle  at  unity  with  the  -j    ,  T  .  ,,  J-  logarithmic 

growth  that  the  ray  to  -j      T  .,    >  makes  with  the  ray  to  unity. 

Plot  on  the  Argand  diagram  two  numbers,  and  the  power  of 
each  of  them  by  the  other. 

135.  Any  number  e{a  +  ib)  is  completely  determined  when  we 
know  the  tensor  <?,  =  r  say,  together  with  b  giving  the  sort. 
For  brevity,  we  write  then  ea  +  ib  =  rb:  so  that  ib,  when  b  grows 
from  o  to  27r,  becomes  in  succession  all  complex  units. 

Just  as  x  -\-  iy  has  its  form  of  growth  determined  by  a 
relation  between  x  and  y,  so  rb  has  its  form  of  growth  deter- 
mined by  a  relation  between  r  and  b. 

If  b  is  constant,  then  rs  is  of  a  constant  sort,  and  the  growth 
is  a  uniform  one;  the  same,  in  fact,  as  that  of  x-\-iy  for 
x  =  ky,  where  k  is  the  ratio  of  the  non-z"  to  the  i  part  of  rb. 
If,  on  the  other  hand,  r  is  constant,  the  growth  is  the  same  as 
that  of  x  -\-  iy  when  x2  -f- y2  =  r2. 

Suppose  r  =  eb.    Then  the  rate  of  growth  of  r  compared  to  b 

eh  —  eb 
is  -v 7-  =  eh  —  r.     But  always  the  b  growth  is  z-ward  to  r 

growth ;  consequently  the  rate  of  growth  of  rb  compared  to  b 
is  rb(i  -\~i),  or  the  logarithmic  rate  of  growth  is  1  — |—  I.  We 
thus  get  one  of  the  logarithmic  spirals  of  §  120. 

Plot  the  growths  of  rb  for  r  =  b,  r  =  b2,  r  —  \/by  r  =  ~. 


TENSOR  REPRESENTATION :   SINES  AND   COSINES.     1 15 
If  \b  =  p  -)-  qi,  show  the  following  equalities  to  be  true  : 

p  —  qi—\_hy      —  p  —  qi=ii  +  2„=ii)_2ir,     j>  +  gi  =  l6  +  2n7r 

where  n  is  integral, 

q  —  ip  =  1  b  -  Tr,  —  q  —  ip  =  lb  -  n,  i=  1*,  —  1  =  I*,  —  i—  i_». 

2  2 

If  also  lb'  =  p'  +  *y,  show  that 

iiXiy=u+*',  i*-*-  i^=i*-yi  PP'—qq'+{pqfJrP'q)  i—  i*+*. 

Express  in  terms  of/,  ^,/',  ^',  the  numbers  i2b,  1^,  iA 

From  §  126  we  have 

1  JL_  —  0.999998823449  4-  0.0015339803/. 

2048 

Calculate  1  w  ,  1^,  I  ,  ,  ly. 

1024     2048     4096     4096 

In  the  calculation  of  §  126,   in   order  to  get  ij^,  we  had 

2048 
first  to  get  the  non-z  parts  of  in,  i„,  iff,  .  .  .     Had  we  at  the 

4        8        76 

same  time  gotten  the  /-parts,  we  could  then  easily  obtain,  by 
the  formulae  immediately  above,  p  -\-  iq  expressions  of  ib  for 

all  values  of  b>  exact  multiples  of ~ . 

VI.    Tensor  Representation:  Sines  and  Cosines. 
136.  Suppose  that  for  1     *m  0  <  1),  on  a  growth  with  con- 

+  2048 

stant  unit  tensor  from  ib  to  I     _*  ,  we  substitute  a  number  on 

W48 


the  uniform  growth  from  ib  to  1     _*_  ,  and  the  same  part  of  the 

2048 

way  on   that   growth  that   li+-  fa  Qn   ^  varying  h_ 


2*48 

What  is  the  error  in  tensor  and  sort  ? 


If  \b  =  p  -j-  ^  and   i^    jr_  =  /'  4"  *?'»  while  /«  +  «=  I,  the 
2048 
number  on  the  uniform  growth  is  mp  -\-  np'  -\-  i{mq  -\-  nq').    Of 
this  the  tensor  squared  is 

m2  +n2  +  2mn(pp'  +  qq')  =  {in  4-  rif  —  2mn(\  —  pp'  —  qq1) 

=  1  —  mn(pf  —  p2  4-  q'  —  q2)- 


Il6       AN  INTRODUCTION   TO    THE   LOGIC   OF  ALGEBRA. 


Here  V(p'  — p)2-\-{q' — ^)2does  not  differ  (§  126)  0.0000000004 

from -.     For  each  we  will  for  brevity  write  h.     Thus  the 

2048  J 

square  of  the  tensor  of  the  substituted  number  is  1  —  mnh2. 
Now  mn  is  never  larger  than  \,  its  value  when  m  =  n  =  -J.  Any- 
other  value  could,  in  fact,  be  written  {i~\-  &)(i  —  k)  =  ^  —  k2. 

_  /z2 
Consequently  mnh2  —  — ,  and*the  error  in  tensor, 


h2 


I  —  Vl  —  /^/z2,  <  —  <  0.000000 1. 
4 

Let  the  substituted  number  grow,  keeping  the  sort  con- 
stant, until  the  tensor  is  unity.  Suppose  that  the  number  then 
becomes  ib+t.     The  error  in  sort  is  t  —-  nh.     We  have 

mp  +  tip'  .    mq  4-  nqf  ,  mmu 

V  I  —  7/z?z/r  y  1  —  mn/12 

Hence  ?  =  (  **  +  &    _,)'+  (j^±M=  _    V 

\ri  -  mnh2         )       \V  \  —  mnh2         / 

2  —  nh2 


Vi  —  mnh2 

2  —  nh2 
Since  — =  >  2  —  ;z>£2,  k2  >  ?z>£2  and  a  fortiori  k2  >  /z2^2. 

y  1  —  ;/z;z/z2 

2 nh^  w /» 

On  the  other  hand,  2  —  =■  < 7-.     For  the 

Vi  —  ;^2      1  —  mnh2 

truth  of  this  last  involves  and  is  involved  in 

n2h2     \2  j,  (2  —  ^//2)2 
mnh 


2 


I  —  mnh2)    ^  1  — 


2 1 


which  after  expansion  and  reduction  becomes  /z2  <  1  —  mnh2t 
and  is  true  because  n2  <  1  —  w*  <  1  —  w«Aa. 


TENSOR  REPRESENTATION :   SINES  AND   COSINES.     117 

Therefore  ■=  >  t  >  nh.     But  because  Vi  —  mnh2 

Vi  —  mnh2 

11J1 

does  not  differ  mnh2  from  1,  does  not  differ  mnh2X 

V I  —  mnh2 

nh  =  rnn*M  from  nh,  and  the  error  in  sort  is  not  as  much  as 

0.000000000 1. 

Had  we  the  numbers/  -\-iq  for  intervals  in  the  value  of  b 

7t 

less  than - ,  the  above  method  of  approximation  to  inter- 

2040 

mediate  p  -j-  iq  numbers  would  give  still  better  results. 

137.  The  numbers  p   and  q,  depending  on  b,  are  called 

respectively  the  cosine  and  the  sine  of  b,  so  that 

cos  b  -f-  1 sin  b  =  ib. 

The  number  b  is  usually  given  for  the  intervals  n  +  180, 
called  degrees,  ir-r-  180-J-60,  called  minutes,  and  ^-r- 180-^-60-^60, 
called  seconds.     Thus : 

one  degree  =  i°  =  0.0174533  .  .  .  , 

one  minute  =  i'  =  0.0002909  .  .  .  , 

one  second  =  \"  —  0.0000048  .  .  .  , 

while  unity  =  $7°  lf  44"-8. 

Comparing  §  135,  write  in  terms  of  the  sines  and  cosines  of 

b  and  b',  cos  (b+  ^),  sin  (b  +^),  cos  (b  +  *),  sin  (b  +  n), 

cos  (b  ±  b'),  sin  {b  ±  b'),  cos  \b,  sin  \b,  cos  2b,  sin  2b,  sin  lb, 

sin  ( b),  cos  (—  b). 

Prove  that  (cos  b  +  i  sin  £)m  =  cos  mb  -f-  f  sin  mb, 
cos  3  =  J(^  +  r%  sin  3  =  l-{e-ib  -  eib). 
Show  that  a  table  giving  the  sine  and  cosine  of  b  for  all 

values  of  b  from  zero  to  -  is  sufficient  for  getting  the  sines  and 

4 
cosines  of  any  value  of  b. 


1 18      AN  INTRODUCTION   TO    THE  LOGIC  OF  ALGEBRA. 

With  such  a  table  let  the  student  find  sin  3,  sin  1.15,  cos  f  ; 
numbers  whose  sines  are  2,  4J-,  —  3,  —  13. 

He  can  roughly  verify  his  values  by  plotting  on  the  Argand 
diagram  the  values  of  ib,  given  or  obtained,  together  with  the 
corresponding/  -j-  iq  numbers. 

138.  Cast  a  glance  back  over  the  route  by  which  we  have 
come. 

We  started  with  familiar  ideas  of  counting,  and  followed 
them  by  the  scarcely  less  familiar  ideas  of  addition,  subtrac- 
tion, multiplication,  division,  involution, and  evolution;  adding 
to  these,  for  the  sake  of  completeness,  the  process  of  taking  a 
logarithm. 

To  make  subtraction  always  possible,  negative  numbers 
were  introduced  and  defined,  and  the  necessary  extensions  of 
the  algebraic  processes  carried  out. 

In  the  same  way,  fractions  made  division  always  possible ; 
while  for  root  extraction  and  taking  of  logarithms,  incommen- 
surables  and  double  numbers  were  needed. 

In  all  of  our  extensions  of  meanings  it  has  been  found 
possible  to  adhere  to  these  rules : 

The  extended  meaning  has  included  the  unextended  as  a 
special  case. 

The  relation  of  process  and  inverse  has  been  maintained. 

The  commutative,  associative,  distributive,  and  index  laws 
that  held  with  the  first  simplest  numbers  have  been  made  to 
hold  throughout. 

Finally,  by  our  extensions,  we  have  made  the  three  primary 
operations  and  their  four  inverses  always  possible,  i.e.  always 
resulting  in  a  number  belonging  to  our  scheme. 

Notice  now  whither  further  progress  may  lead  us. 

In  the  reasoning  first  employed  for  the  handling  of  incom- 
mensurables  and  further  developed  in  the  treatment  of  growths 
and  rates  we  have  the  germ  of  that  marvellous  invention  of 
Leibnitz  and  Newton,  the  Infinitesimal  Calculus. 

The  graphic  representation  leads  to  the  Analytic  Geometry 
of  Des  Cartes. 

The  theory  of  sines  and  cosines  with  its  geometric  applica- 


TENSOR  REPRESENTATION :    SINES  AND   COSINES.     119 

tions  is  Trigonometry  and  leads,  by  the  introduction  of  new 
numbers  and  conceptions,  to  the  Function  Theory. 

The  theory  of  double  numbers  is  simple  and  restricted,  and 
but  a  faint  suggestion  of  what  is  to  be  found  in  the  beautiful 
developments  proposed  by  Hamilton  and  Grassmann,  the 
Peirces  and  Sylvester.  These  are  a  few  only  of  the  lines  of 
thought  open  to  the  student. 

Yet,  in  whatsoever  direction  investigation  may  carry  him, 
he  will  find  his  work  essentially  the  same  in  character.  Defini- 
tions and  conventions  and  their  logical  consequences  and  rela- 
tions make  up  the  whole  of  it.  These  relations  form  the  uni- 
verse wherein  the  Mathematician  lives ;  a  universe,  to  be  sure, 
of  his  own  construction,  a  product  of  his  brain,  but  none  the 
less  real  and  substantial  to  him.  Here  he  observes  and  com- 
pares and  experiments ;  here  he  reasons  out  connections,  dis- 
covers causes,  and  foretells  results. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN     INITIAL     FINE     OF    25     CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


Mi-  Itfof 


JUL   211933 


DK  23  1933 


Mtft 


31941 


2Apr»60BSf 
LIBRARY  USE 

MAR  19 1960 

REC'D  LD 

MAfl  i  $  i960 


NOV    8  1941 


NOV   18  1942 


SfiObf 


«?$ 


O  LD 


°ct  is  1946   OCT  1 4 


JAN 


3®  '*M1Mfflo  5 


SApr'55  J  P 


B  19 


re 


0  L 


962 


2r, 


.  .  LD  21-SOm-l, 


